estimation for self-adjoint eigenvalue problems Liu Xuefeng Research - PowerPoint PPT Presentation

. .. .. . . .. . . . . . .. . .. .. . . .. .. . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan Proof: . . Theorem (Upper eigenvalue bounds) . . . . Upper eigenvalue bounds Error constant in Rayleigh quotient form . .. . . . . . . . . . . .. . . .. . . .. . . .. .. . .. .. . . .. . . .. 9 / 79 . . . . .. . . .. Eigenvalue problem in V h Let ( λ h,k , φ h,k ) k =1 , ··· ,n ( λ h,k ≤ λ h,k +1 ) be the eigen-pairs such that, ∀ v h ∈ V h . M ( v h , φ h,k ) = λ h,k N ( v h , φ h,k ) If V h ⊂ V , then an upper bound for λ k is given as, λ k ≤ λ h,k . Let E h,k be the space spanned by { φ h, 1 , · · · , φ h,k } . Then, λ k ≤ max R ( u ) = λ h,k . u ∈ E h,k 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. .. .. . . .. . . . . . .. . .. .. . . .. .. . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan not easy to construct. . . Theorem (Upper eigenvalue bounds) . . . . Upper eigenvalue bounds Error constant in Rayleigh quotient form . .. . . . . . . . . . . .. . . .. . . .. . . .. .. . .. .. . . .. . . .. 9 / 79 . . . . .. . . .. Eigenvalue problem in V h Let ( λ h,k , φ h,k ) k =1 , ··· ,n ( λ h,k ≤ λ h,k +1 ) be the eigen-pairs such that, ∀ v h ∈ V h . M ( v h , φ h,k ) = λ h,k N ( v h , φ h,k ) If V h ⊂ V , then an upper bound for λ k is given as, λ k ≤ λ h,k . In the field of finite element method (FEM), a V h satsifying V h ⊂ V is called conforming FEM space ( 適合有限要素空間 ) . For eigenvalue problem associated with high-order derivatives, V h ( ⊂ V ) is 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . .. .. . .. . . .. . . .. . . .. . .. . . . .. . . .. . Error constant in Rayleigh quotient form Theorem for lower eigenvalue bounds Theorem 1 . . Assertion: The lower bounds for eigenvalues are given as, Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . .. .. . . .. . . .. . . .. . . . . . .. . . . .. . . . .. 10 / 79 . .. .. . . .. . . Let P h : V → V h be a projection satisfying for all v h ∈ V h M ( u − P h u, v h ) = 0 , Moreover, suppose that an error estimation for P h is given as, | u − P h u | N ≤ C h | u − P h u | M . λ h,k /(1 + λ h,k C 2 h ) ≤ λ k ( k = 1 , 2 , · · · , n ) . 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . .. . . .. . . .. . Error constant in Rayleigh quotient form .. . . .. . . .. . Main proof for Theorem 1 . . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan max , we have, With the error estimation for . . About max With current assumption, the min-max principle holds for eigenvalues: max the following inequality holds, )=k, . By showing that dim( be the space spanned by Let max min . .. .. . . . .. . . .. . .. .. . . .. . . .. . . . 11 / 79 . . . .. . . .. . . .. . . .. . . .. . . .. λ h,k = R ( u ) u ∈ H dim ( H )= k ; H ⊂ V h 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. .. .. . . .. . . . . . .. . . .. . . . .. . About Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan max , we have, With the error estimation for . . . . max the following inequality holds, max min With current assumption, the min-max principle holds for eigenvalues: Main proof for Theorem 1 Error constant in Rayleigh quotient form .. .. . .. .. . .. .. . . . . . .. . . .. . . . . .. . . . .. . . .. . . .. . 11 / 79 .. . . . .. . λ h,k = R ( u ) u ∈ H dim ( H )= k ; H ⊂ V h Let E k be the space spanned by { φ 1 , · · · , φ k } . By showing that dim( P h E k )=k, λ h,k ≤ v h ∈ P h E k R ( v h ) = max v ∈ E k R ( P h v ) . 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . . . .. . . .. . . .. .. . .. . . .. .. . . . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan max , we have, With the error estimation for . . max .. the following inequality holds, max min With current assumption, the min-max principle holds for eigenvalues: Main proof for Theorem 1 Error constant in Rayleigh quotient form . . . .. .. .. . . .. . . . . . .. . . .. . . . . .. . . .. . . .. . . . .. 11 / 79 . . .. . . .. λ h,k = R ( u ) u ∈ H dim ( H )= k ; H ⊂ V h Let E k be the space spanned by { φ 1 , · · · , φ k } . By showing that dim( P h E k )=k, λ h,k ≤ v h ∈ P h E k R ( v h ) = max v ∈ E k R ( P h v ) . About R ( P h v ) R ( P h v ) = | P h v | 2 M | P h v | 2 N 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . . . .. . . .. .. . .. . . .. . . .. .. . . . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan max , we have, With the error estimation for . . max .. the following inequality holds, max min With current assumption, the min-max principle holds for eigenvalues: Main proof for Theorem 1 Error constant in Rayleigh quotient form . . . .. .. .. . . .. . . . . . .. . . .. . . . . .. . . .. . . .. . . .. . 11 / 79 .. . .. . . . λ h,k = R ( u ) u ∈ H dim ( H )= k ; H ⊂ V h Let E k be the space spanned by { φ 1 , · · · , φ k } . By showing that dim( P h E k )=k, λ h,k ≤ v h ∈ P h E k R ( v h ) = max v ∈ E k R ( P h v ) . About R ( P h v ) R ( P h v ) = | P h v | 2 = | v | 2 M − | v − P h v | 2 M M | P h v | 2 | P h v | 2 N N 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . . . .. . .. .. . . .. . . .. . . .. .. . . . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan max , we have, With the error estimation for . . max .. the following inequality holds, max min With current assumption, the min-max principle holds for eigenvalues: Main proof for Theorem 1 Error constant in Rayleigh quotient form . . . .. .. . . .. . . . . . . .. . . .. . . .. 11 / 79 . . . .. . . .. .. . .. . . . .. . .. . . λ h,k = R ( u ) u ∈ H dim ( H )= k ; H ⊂ V h Let E k be the space spanned by { φ 1 , · · · , φ k } . By showing that dim( P h E k )=k, λ h,k ≤ v h ∈ P h E k R ( v h ) = max v ∈ E k R ( P h v ) . About R ( P h v ) R ( P h v ) = | P h v | 2 = | v | 2 M − | v − P h v | 2 | v | 2 M − | v − P h v | 2 M M M ≤ | P h v | 2 | P h v | 2 ( | v | N − | v − P h v | N ) 2 N N 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. .. .. . . .. . . . . . .. . . .. . . .. .. the following inequality holds, Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan max . . . max max . min With current assumption, the min-max principle holds for eigenvalues: Main proof for Theorem 1 Error constant in Rayleigh quotient form . .. . . .. . . . . .. . . .. . . .. . . .. . . .. . . . .. . .. .. . . .. . 11 / 79 . . . .. . .. . λ h,k = R ( u ) u ∈ H dim ( H )= k ; H ⊂ V h Let E k be the space spanned by { φ 1 , · · · , φ k } . By showing that dim( P h E k )=k, λ h,k ≤ v h ∈ P h E k R ( v h ) = max v ∈ E k R ( P h v ) . About R ( P h v ) R ( P h v ) = | P h v | 2 = | v | 2 M − | v − P h v | 2 | v | 2 M − | v − P h v | 2 M M M ≤ | P h v | 2 | P h v | 2 ( | v | N − | v − P h v | N ) 2 N N With the error estimation for P h , we have, λ k R ( P h v ) ≤ . 1 − C 2 h λ k 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. .. .. . . .. . . . . . .. . . .. . . .. .. the following inequality holds, Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan max . . . max max . min With current assumption, the min-max principle holds for eigenvalues: Main proof for Theorem 1 Error constant in Rayleigh quotient form . .. . . .. . . . . .. . . .. . . .. . . .. . . .. . . . .. . .. .. . . .. . 11 / 79 . . . .. . .. . λ h,k = R ( u ) u ∈ H dim ( H )= k ; H ⊂ V h Let E k be the space spanned by { φ 1 , · · · , φ k } . By showing that dim( P h E k )=k, λ h,k ≤ v h ∈ P h E k R ( v h ) = max v ∈ E k R ( P h v ) . About R ( P h v ) R ( P h v ) = | P h v | 2 = | v | 2 M − | v − P h v | 2 | v | 2 M − | v − P h v | 2 M M M ≤ | P h v | 2 | P h v | 2 ( | v | N − | v − P h v | N ) 2 N N With the error estimation for P h , we have, λ k v ∈ E k R ( P h v ) ≤ . 1 − C 2 h λ k 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. .. .. . . .. . . . . . .. . . .. . .. .. .. the following inequality holds, Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan max . . . max max . min With current assumption, the min-max principle holds for eigenvalues: Main proof for Theorem 1 Error constant in Rayleigh quotient form . .. . . . . . . . .. . . .. . . . . . .. . . .. .. . . .. . . .. . . .. .. 11 / 79 . . . .. . .. . λ h,k = R ( u ) u ∈ H dim ( H )= k ; H ⊂ V h Let E k be the space spanned by { φ 1 , · · · , φ k } . By showing that dim( P h E k )=k, λ k λ h,k ≤ v h ∈ P h E k R ( v h ) = max v ∈ E k R ( P h v ) ≤ . 1 − C 2 h λ k About R ( P h v ) R ( P h v ) = | P h v | 2 = | v | 2 M − | v − P h v | 2 | v | 2 M − | v − P h v | 2 M M M ≤ | P h v | 2 | P h v | 2 ( | v | N − | v − P h v | N ) 2 N N With the error estimation for P h , we have, λ k v ∈ E k R ( P h v ) ≤ . 1 − C 2 h λ k 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. .. . .. . . .. . . .. . . .. . . . .. . .. . . .. . . .. . Error constant in Rayleigh quotient form Inspired by the result in X. Liu and S. Oishi. Verified eigenvalue evaluation for the Laplacian over polygonal domains of arbitrary shape. SIAM J. Numer. Anal., 51(3):1634–1654, 2013 . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . .. . . . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. 12 / 79 (Only conforming FEM space V h ( ⊂ V ) is considered.) 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . . .. . . .. . .. .. . . .. . . .. . . . that is also a projection operator Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan , of triangulation That is, on each element . will be a good candidate for A locally defined interpolation operator . . . Two tasks in the application of Theorem 1 . Error constant in Rayleigh quotient form . .. . .. .. .. . . . .. . . . . . .. . . .. . . . .. .. . . .. . . .. . . .. 13 / 79 . . .. . . .. . 1) Selection of proper FEM space V h and the projection P h : for all v h ∈ V h . M ( u − P h u, v h ) = 0 , 2) Explicit error estimation for P h : | u − P h u | N ≤ C h | u − P h u | M . 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . . .. .. . .. . . .. . . .. . . .. . .. .. . . .. . . .. . Error constant in Rayleigh quotient form . Two tasks in the application of Theorem 1 . . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . . .. . . .. . . .. . . .. . . .. . . . . . .. . . .. . . .. . .. . 13 / 79 . .. . 1) Selection of proper FEM space V h and the projection P h : for all v h ∈ V h . M ( u − P h u, v h ) = 0 , 2) Explicit error estimation for P h : | u − P h u | N ≤ C h | u − P h u | M . A locally defined interpolation operator Π h that is also a projection operator will be a good candidate for P h . That is, on each element K of triangulation T h , ( P h u ) | K = Π h ( u | K ) . 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . .. . . .. . . .. . . .. . . .. . .. . . . .. . . .. . Error constant in Rayleigh quotient form . . 2. Eigenvalue problem of Laplace operator Two kind of projection ’s will be considered. Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . .. .. .. . . .. . . .. . . .. . . .. . . . . . . .. . . .. . . .. . . .. . . .. 14 / 79 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. .. . .. . . .. . . .. . . .. . . . .. . .. . . .. . . .. . Error constant in Rayleigh quotient form . . 2. Eigenvalue problem of Laplace operator Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . .. . . . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. 14 / 79 Two kind of projection P h ’s will be considered. 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. .. .. . . .. . . . . . .. . . .. .. . .. .. . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . Definition of operators: (2D) . . Eigenvalue problem for 2rd order differential operator . . Eigenvalue problem of Laplace operators Error constant in Rayleigh quotient form . .. . . . . . . . .. . . .. . . .. . . .. . . .. . . .. .. . . .. . .. . 15 / 79 . . .. . . .. . . Assumption: Ω is a simply connected bounded domain. − ∆ u = λu, u = 0 on ∂ Ω , ∇ u = ( u x , u y ) for u being a scalar function. div p = p 1 ,x + p 2 ,y for p = ( p 1 , p 2 ) being a vector function. ∆ u = div · ∇ u = u xx + u yy . 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. .. .. . . .. . . . . . .. .. . .. . . .. .. . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . Variational formulation: . . Eigenvalue problem for 2rd order differential operator . . Eigenvalue problem of Laplace operators Error constant in Rayleigh quotient form . .. . . . . . . . .. . . .. . . .. . . .. . . .. . . .. .. . . .. .. . . . . . . . .. 15 / 79 .. . Assumption: Ω is a simply connected bounded domain. − ∆ u = λu, u = 0 on ∂ Ω , Let V := { v ∈ H 1 (Ω) | u = 0 on ∂ Ω } . ∫ ∫ Find u ∈ V and λ ≥ 0 s.t. ∇ u · ∇ vdx = λ uvdx ∀ v ∈ V . Ω Ω 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. .. .. . . .. . . . . .. .. . . .. . . .. .. . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . Variational formulation: . . Eigenvalue problem for 2rd order differential operator . . Eigenvalue problem of Laplace operators Error constant in Rayleigh quotient form . .. . . . . . . . .. . . . . . .. . . .. . . .. .. . .. .. . .. .. . . . . . . . . .. 15 / 79 . .. Assumption: Ω is a simply connected bounded domain. − ∆ u = λu, u = 0 on ∂ Ω , Let V := { v ∈ H 1 (Ω) | u = 0 on ∂ Ω } . Find u ∈ V and λ ≥ 0 s.t. M ( u, v ) = λN ( u, v ) ∀ v ∈ V . where M ( · , · ) and N ( · , · ) are bilinear forms over V : ∫ ∫ M ( u, v ) = ∇ u · ∇ vdx, N ( u, v ) = uvdx . Ω Ω 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. .. . .. . . .. . . .. . . .. . . . .. . .. . . .. . . .. . Error constant in Rayleigh quotient form . . Conforming FEM space Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . .. . . . .. . . .. . . .. . . .. . . .. . . .. . . .. . .. . . . .. . . .. 16 / 79 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . . .. . . .. . . .. . . .. . . .. .. .. .. . . .. . . .. . Error constant in Rayleigh quotient form . . . Base function Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . . . . . . .. . . .. . . .. . . .. . 17 / 79 .. .. .. . .. . . . . . . .. . . .. . Eigenvalue problem in FEM spaces V h Lagrange FEM space: V h ( ⊂ V ) Let T h be a triangulation of domain Ω . The function space V h over T h is consisted of function v h such that, 1) v h | K is linear function on each element K ∈ T h ; 2) v h is a continuous function over Ω . 0.14 0.12 0.14 0.1 0.12 0.08 0.06 0.1 0.04 0.08 0.02 0.06 0 0.04 0.02 0 0 0.2 0.4 0.6 1 0.8 0.8 0.6 0.4 0.2 1 0 Triangulation T h of domain Sample function u h ∈ V h 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . .. . . .. . . .. . . .. . . .. . .. . . . .. . . .. . Error constant in Rayleigh quotient form . . . . . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . .. .. .. . . .. . . .. . . .. . . .. . . . . . . .. . . . .. 17 / 79 . .. . . .. . . .. Eigenvalue problem in FEM spaces V h Lagrange FEM space: V h ( ⊂ V ) Let T h be a triangulation of domain Ω . The function space V h over T h is consisted of function v h such that, 1) v h | K is linear function on each element K ∈ T h ; 2) v h is a continuous function over Ω . The bilinear forms M ( · , · ) and N ( · , · ) over V h : ∫ ∫ M ( u h , v h ) = ∇ u h · ∇ v h dx, N ( u h , v h ) = u h v h dx . Ω Ω Eigenvalue problem in V h : Find u h ∈ V h and λ h ≥ 0 s.t. ∀ v h ∈ V h . M ( u h , v h ) = λ h N ( u h , v h ) 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. .. .. . . .. . . . . . .. .. . .. . . .. .. problems below, respectively, Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan domains of arbitrary shape. SIAM J. Numer. Anal., 51(3):1634–1654, 2013 , X. Liu and S. Oishi, Verified eigenvalue evaluation for the Laplacian over polygonal shape. (3) We have an error estimate as below, . . . . Error constant in Rayleigh quotient form . .. . . . . . . . .. . . .. . . .. . . .. . . .. . . . .. . . .. . . .. .. 18 / 79 . . .. . . .. . Error estimation for P h : V → V h A priori error estimation ( 事前誤差評価 ) Given f ∈ L 2 (Ω) , let u ∈ H 1 0 (Ω) and P h u ∈ V h be the solutions of variational for v h ∈ V h . for v ∈ H 1 M ( u, v ) = N ( f, v ) 0 (Ω) , M ( P h u, v h ) = N ( f, v h ) ∥ u − P h u ∥ L 2 ≤ C h ∥∇ ( u − P h u ) ∥ L 2 ≤ C 2 h ∥ f ∥ L 2 . where C h is a quantity only depending on the mesh triangulation and domain 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . Error constant in Rayleigh quotient form Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . .. .. . . . .. . . .. . . .. . . . . .. .. . . .. . . .. . . .. . . .. . 19 / 79 Remarks on evaluation of C h For convex domain, the solution u corresponding for given f is regular, that is, u ∈ H 2 (Ω) . The constant C h is easy to obtain. For non-convex domain, u may have a singularity, that is, u ̸∈ H 2 (Ω) . A new “Hypercircle equation method” is developed to give an estimation of C h . 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . .. . . .. .. . .. . . .. . . .. . .. . . . .. . . .. . Error constant in Rayleigh quotient form Lower eigenvalue bounds based on Theorem 1 . Setting for application of Theorem 1 . . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . .. .. . . .. . . .. . . . . . .. . . .. . . . . .. . . .. 20 / 79 . .. . . .. . . .. V = H 1 0 (Ω) ; V h : Lagrange conforming FEM space ( V h ⊂ V ); ∫ M ( u, v ) := Ω ∇ u · ∇ vdx ; ∫ N ( u, v ) := Ω uvdx ; Projection P h : M ( u − P h u, v h ) = 0 for v h ∈ V h . Error estimation for P h : | u − P h u | N ≤ C h | u − P h u | M 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. .. . .. . . .. . . .. . . .. . . . .. . .. . . .. . . .. . Error constant in Rayleigh quotient form . . Computation results based on conforming FEMs Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . .. . . . .. . . .. . . .. . . .. . . .. . . .. . . .. . .. . . . .. . . .. 21 / 79 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . upper lower Exact values: Example I: unit isosceles right triangle domain Error constant in Rayleigh quotient form . .. . .. 1 . .. .. . . .. . . rel. err 48.9488 . 153.8131 Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan Lower and upper bounds of eigenvalues 0.15888 201.5760 171.9199 5 0.1018 170.3116 4 49.5525 0.06239 129.7290 121.8806 3 0.03238 99.6328 96.4497 2 0.01225 .. . . . .. . . .. . . .. . . .. . .. .. . . .. . . .. . . .. . . . 22 / 79 .. . . . .. . . .. . . . .. . .. . . Function space: V = H 1 ∥ u ∥ 2 0 (Ω) , V = ( ∇ u, ∇ u ) L 2 (Ω) . Problem: Find u ∈ H 1 0 (Ω) and λ > 0 such that, − ∆ u = λu in Ω , u = 0 on Γ . Constant values: h = 1/32 , C 1 ≤ 0 . 493 , Thus M := 0 . 0155 λ 1 = 5 π 2 ≈ 49 . 348 , λ 2 = 10 π 2 ≈ 98 . 696 , λ 3 = 13 π 2 ≈ 128 . 30 λ i (0 , 1) (0 , 0) (1 , 0) 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . rel. err. upper lower Example II: Square with hole Error constant in Rayleigh quotient form . .. . .. 9.611519 . . .. . . .. .. . 1 10.088400 . 11.010400 Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan First eigenfunction and mesh Figure : 0.084110 16.405400 15.081232 5 0.052318 10.449044 0.048414 4 0.050115 10.487100 9.974390 3 0.050150 10.486400 9.973375 2 .. . . . . . .. . . .. . . .. . . .. . . .. . . .. . . .. .. . . . . .. . . .. . . .. .. . .. . . .. . . 23 / 79 Ω : Square (0 , π ) × (0 , π ) excluding hole surrounding by sin 2 ( x ) + sin 2 ( y ) = 3/2 . Exact value: λ 1 = 10 . M := 0.0618 (outter polygon), 0.0634 (inner polygon). λ i Y Z X 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . . .. .. . .. . . .. . . .. . . .. . .. .. . . .. . . .. . Error constant in Rayleigh quotient form Example III: Mixed boundary on square with crack Problem: Table : Eigenvalue estimation (non-uniform mesh) Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . . .. . . .. . . .. . . .. . . .. . . . . . .. . . . . .. .. . .. . . 24 / 79 .. . Let Ω be unit square with crack { ( x, 0 . 5) | 0 < x < 0 . 5 } . ∂ D Ω = ∂ Ω ∩ { y = 1 or y = 0 , or x = 1 } , ∂ N Ω = ∂ Ω/ ∂ D Ω − ∆ u = λu in Ω , u = 0 on ∂ D Ω , ∂u / ∂n = 0 on ∂ N Ω 1 0 0 1 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . upper lower Eigenvalue estimation: (non-uniform mesh) Example III: Mixed boundary on square with crack Error constant in Rayleigh quotient form . .. . .. 1 . . .. . . .. . . relative. 12.233 . 51.049 Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan Eigenfunctions for leading 4 eigenvalues 0.050 71.768 68.241 5 0.037 52.998 4 12.343 0.022 32.119 31.392 3 0.012 16.276 16.087 2 0.009 .. .. . . . .. . . .. . . .. . .. .. . . .. . . .. . . . . .. . . . .. . . .. . . .. . . .. . . .. . . .. 25 / 79 M = 0 . 027 λ i 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. .. . .. . . .. . . .. . . .. . . . .. . .. . . .. . . .. . Error constant in Rayleigh quotient form . . Non-conforming FEM Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . .. . . . .. . . .. . . .. . . .. . . .. . . .. . . .. . .. . . . .. . . .. 26 / 79 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . .. . .. .. . . .. . . .. . . .. . .. . . . .. . . .. . Error constant in Rayleigh quotient form . . . 2) mid-points of interior edges. Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . .. .. . . .. . . .. . . .. . . .. . . . . . . .. . . . . .. .. . .. . 27 / 79 .. . . Eigenvalue problem in FEM spaces V h Crouzeix-Raviart FEM space: V h ( ̸⊂ V ) The function v h of V h satisfies, 1) v h is linear on each element K ∈ T h ; ∫ ∫ e v h ds is continuous on interior edges; e v h ds = 0 on boundary edges; Function v h is only continuous on the 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . .. . . .. . . .. .. . .. . . .. . .. . . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . 2) . . Error constant in Rayleigh quotient form . . .. . . .. . . . .. .. . . .. . . .. . . .. . . .. . . . . . . .. . . . .. . 27 / 79 .. . . .. .. . . Eigenvalue problem in FEM spaces V h Crouzeix-Raviart FEM space: V h ( ̸⊂ V ) The function v h of V h satisfies, 1) v h is linear on each element K ∈ T h ; ∫ ∫ e v h ds is continuous on interior edges; e v h ds = 0 on boundary edges; Extend the bilinear forms M and N from V to V h : ∫ ∫ ∑ M ( u h , v h ) = ∇ u h · ∇ v h dx, N ( u h , v h ) = u h v h dx . Ω K K ∈T h 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. .. . . .. .. . .. . . .. . . .. . . . . . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . . 2) . . . Error constant in Rayleigh quotient form . .. . . .. .. . . . . . .. . .. .. . . .. . . .. . . .. . . . . . .. . .. .. . . . . . .. 27 / 79 .. . Eigenvalue problem in FEM spaces V h Crouzeix-Raviart FEM space: V h ( ̸⊂ V ) The function v h of V h satisfies, 1) v h is linear on each element K ∈ T h ; ∫ ∫ e v h ds is continuous on interior edges; e v h ds = 0 on boundary edges; Eigenvalue problem in V h Find u h ∈ V h and λ h ≥ 0 s.t. ∀ v h ∈ V h . M ( u h , v h ) = λ h N ( u h , v h ) 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . . .. .. . . .. . . .. . . .. . .. . . . .. . . .. . . .. . Error constant in Rayleigh quotient form . . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan .. . . . . . .. . . .. . . .. . . .. . .. .. .. . . .. . . . . . .. . . .. . 28 / 79 The Crouzeix-Raviart interpolation Π h On triangle element K , (Π h u ) | K is a linear function such that ∫ u − Π h u ds = 0 ( i = 1 , 2 , 3) . e i 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . .. . . .. . . .. . .. .. . . .. . .. . . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan Hint: . . . . Error constant in Rayleigh quotient form . . .. . . .. . . . .. .. . . .. . . . . . .. . . .. . . .. . . . .. . . .. . . 28 / 79 . . .. . .. . .. The Crouzeix-Raviart interpolation Π h On triangle element K , (Π h u ) | K is a linear function such that ∫ u − Π h u ds = 0 ( i = 1 , 2 , 3) . e i Important property of Π h For u ∈ H 1 (Ω) , ∫ ∀ v h ∈ P 1 ( K ) . ∇ ( u − Π h u ) · ∇ v h dx = 0 , K ∫ ∫ ∫ ∇ ( u − Π h u ) · ∇ v h dx = ( u − Π h u ) v h ds − ( u − Π h u ) · ∆ v h dx K ∂K K 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. .. . . .. . . .. . .. .. . . .. . . . . . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan Thus, Hence . . . . . Error constant in Rayleigh quotient form . .. . . .. .. . . .. . . .. . . .. . . .. . . .. . . .. . . . . . .. . .. .. . . . . . .. . 28 / 79 .. The Crouzeix-Raviart interpolation Π h On triangle element K , (Π h u ) | K is a linear function such that ∫ u − Π h u ds = 0 ( i = 1 , 2 , 3) . e i Important property of Π h For u ∈ H 1 (Ω) , ∫ ∀ v h ∈ P 1 ( K ) . ∇ ( u − Π h u ) · ∇ v h dx = 0 , K ∫ ∑ ∀ v h ∈ V h ∇ ( u − Π h u ) · ∇ v h dx = 0 , . K K ∈T h 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. .. . . .. . . .. . . .. . . .. . . . . . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan Thus, Hence . . . . . Error constant in Rayleigh quotient form . .. . . .. .. .. . . . . .. . . .. .. . .. . . .. . . .. . . . . . .. .. . .. . . . . . .. . 28 / 79 .. The Crouzeix-Raviart interpolation Π h On triangle element K , (Π h u ) | K is a linear function such that ∫ u − Π h u ds = 0 ( i = 1 , 2 , 3) . e i Important property of Π h For u ∈ H 1 (Ω) , ∫ ∀ v h ∈ P 1 ( K ) . ∇ ( u − Π h u ) · ∇ v h dx = 0 , K ∫ ∑ ∀ v h ∈ V h ∇ ( u − Π h u ) · ∇ v h dx = 0 , . K K ∈T h ∀ v h ∈ V h . M ( u − Π h u, v h ) = 0 , 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . . . .. . . .. . . .. . . .. . . .. .. . . , define Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan sup : over Constant by of .. Over . . Error constant . Error constant in Rayleigh quotient form . . .. .. .. . . . .. . . . . . .. . . .. . . . .. .. . . .. . . .. . . .. . . .. . . .. . 29 / 79 Interpolation error estimation for Π h On each element K , we consider the error estimation by using C e ( K ) : for u ∈ H 1 ( K ) . ∥ u − Π h u ∥ 0 ,K ≤ C e ( K ) | u − Π h u | 1 ,K where | · | d,K denotes the semi-norm of Sobolev function space H d ( K ) . 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . . .. . . .. . .. .. . . .. . . .. . .. .. . . .. . . .. . Error constant in Rayleigh quotient form . . . sup Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . . .. . . .. . . .. . . .. . . .. . . . . . .. . . .. . . 29 / 79 . .. . .. . .. . Interpolation error estimation for Π h On each element K , we consider the error estimation by using C e ( K ) : for u ∈ H 1 ( K ) . ∥ u − Π h u ∥ 0 ,K ≤ C e ( K ) | u − Π h u | 1 ,K where | · | d,K denotes the semi-norm of Sobolev function space H d ( K ) . Error constant C e ( K ) Over K of T h , define V e ( K ) by ∫ V e ( K ) := { v ∈ H 1 ( K ) | vds = 0 , i = 1 , 2 , 3 } . e i Constant C e ( K ) over V e ( K ) : ∥ u ∥ 0 ,K C e ( K ) := | u | 1 ,K , u ∈ V e ( K ) 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . .. . . .. . . .. . . .. . . .. . .. . . . .. . . .. . Error constant in Rayleigh quotient form . . For unit isoceles right triangle , an easy-to-obtain lower bound: Approximate computation shows that, Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . .. .. .. . . .. . . .. . . .. . . .. . . . . . . .. . . . .. 30 / 79 . .. .. . .. . . . Upper bound of C e ( K ) The eigenvalue problem for C e ( K ) : Over K , � ∂u � − ∆ u = λu, = c i ( c i : to be determined ) � ∂n � e i 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. .. . .. . . .. . .. .. . . .. . . . .. . .. . . .. . . .. . Error constant in Rayleigh quotient form . . Approximate computation shows that, Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . . .. . . .. . . .. . . .. . . .. . . . . . .. . . . .. . 30 / 79 .. . . .. . . .. Upper bound of C e ( K ) The eigenvalue problem for C e ( K ) : Over K , � ∂u � − ∆ u = λu, = c i ( c i : to be determined ) � ∂n � e i For unit isoceles right triangle ˆ K , an easy-to-obtain lower bound: K ) ≤ 1 C e ( ˆ π . 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. .. . .. . . .. . . .. . . .. . . . .. . .. . . .. . . .. . Error constant in Rayleigh quotient form . . Approximate computation shows that, Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . .. . .. . . .. . . .. . . .. . . .. . . . . . .. . . . .. . 30 / 79 .. . . .. .. . . Upper bound of C e ( K ) The eigenvalue problem for C e ( K ) : Over K , � ∂u � − ∆ u = λu, = c i ( c i : to be determined ) � ∂n � e i For unit isoceles right triangle ˆ K , an easy-to-obtain lower bound: K ) ≤ 1 C e ( ˆ π . K ) ≈ 0 . 2377 < 1 C e ( ˆ π ≈ 0 . 3183 . 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . . .. . . .. . . .. .. . .. . . .. . .. .. . . .. . . .. . Error constant in Rayleigh quotient form . . . Hence, Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . . . . . .. . . . .. . . .. . . .. . 31 / 79 .. . . .. . . .. . . .. . . .. .. . . Error estimation for projection Π h Upper bound for C e ( K ) For general triangle K with the maximum inner angle as θ and the second longest edge length as L , C e ( K ) ≤ L √ 1 + | cos θ | π ∥ u − Π h u ∥ 0 ,K ≤ L √ 1 + | cos θ | | u − Π h u | 1 ,K π B { T αL θ A { O L 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . . . .. . . .. . . .. . . .. . .. . . . .. . . .. . . .. . Error constant in Rayleigh quotient form singularities. Easy to deal with boundary conditions. Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan .. .. . . . . .. . . .. . . .. . . .. .. . .. .. . . .. . . . . . .. . . .. . 32 / 79 Remarks on Crouzeix-Raviart interpolation Π h Even for non-convex domains, the interpolation Π h is well-defined for u ∈ H 1 (Ω) . Thus there is no additional efforts needed for problems with 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . .. . . .. . . .. . . .. . . .. . .. . . . .. . . .. . Error constant in Rayleigh quotient form Lower eigenvalue bounds based on Theorem 1 . Setting for application of Theorem 1 . . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . .. .. .. . . .. . . .. . . .. . . .. . . . . . . . .. . . .. 33 / 79 . .. . . .. . . .. V = H 1 0 (Ω) ; V h : Crouzeix-Rarviart FEM space ( V h ̸⊂ V ); M ( u, v ) := ∑ ∫ K ∇ u · ∇ vdx ; K ∈T h ∫ N ( u, v ) := Ω uvdx ; Projection P h := Π h : M ( u − P h u, v h ) = 0 for v h ∈ V h . Error estimation for P h : ( ) | u − P h u | N ≤ C h | u − P h u | M C h := max K ∈T h C e ( K ) 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. .. . .. . . .. . . .. . . .. . . . .. . .. . . .. . . .. . Error constant in Rayleigh quotient form . . General Crouzeix-Raviart interpolation operators Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . .. . . . .. . . .. . . .. . . .. . . .. . . .. . . .. . .. . . . .. . . .. 34 / 79 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . . . .. . . .. . . .. . . .. .. . .. . . . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan , For any . . Property Extension of Crouzeix-Raviart interpolation in 3D .. Crouzeix-Raviart type interpolation in 3D Error constant in Rayleigh quotient form . .. . . .. . . . . .. .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . 35 / 79 .. . . . .. . . Let K be a tetrahedron with surfaces S i , i = 1 , 2 , 3 , 4 . For u ∈ H 1 ( K ) , Π h u ∈ P 1 ( K ) is determined by ∫ ( u − Π h u ) dS = 0 , i = 1 , 2 , 3 . S i 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . .. . . .. . . .. . . .. . . .. . .. . Crouzeix-Raviart type interpolation in 3D Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . Property . Extension of Crouzeix-Raviart interpolation in 3D Error constant in Rayleigh quotient form . . .. . . .. . . .. .. .. . . .. . . .. . . .. . . .. . . . . . . .. . . .. . 35 / 79 . . .. . . .. .. . Let K be a tetrahedron with surfaces S i , i = 1 , 2 , 3 , 4 . For u ∈ H 1 ( K ) , Π h u ∈ P 1 ( K ) is determined by ∫ ( u − Π h u ) dS = 0 , i = 1 , 2 , 3 . S i For any v ∈ P 1 ( K ) , ∫ ∇ ( u − Π h u ) · ∇ vdX = 0 . K 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . . .. . . .. . .. .. . . .. . . .. . . . . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan , For any . . Property . . . Extension of Crouzeix-Raviart interpolation in 1D . Crouzeix-Raviart type interpolation in 1D Error constant in Rayleigh quotient form . .. .. . .. .. . . . .. . . . . . .. . . .. . . . .. .. . . .. . . .. . . 36 / 79 .. . . .. . . .. . Let I be an interval with two vertices a and b . For u ∈ H 1 ( I )( ⊂ C 0 ( I )) , Π h u ∈ P 1 ( I ) is determined by (Π h u )( a ) = u ( a ) , (Π h u )( b ) = u ( b ) . 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. .. .. . . .. . . . . . .. . . .. . . .. .. . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . Property . . Extension of Crouzeix-Raviart interpolation in 1D . . Crouzeix-Raviart type interpolation in 1D Error constant in Rayleigh quotient form . .. . . .. . . . . .. . . .. . . .. . . .. . . .. . . .. .. . . .. . .. . 36 / 79 . . . . . .. . .. Let I be an interval with two vertices a and b . For u ∈ H 1 ( I )( ⊂ C 0 ( I )) , Π h u ∈ P 1 ( I ) is determined by (Π h u )( a ) = u ( a ) , (Π h u )( b ) = u ( b ) . For any v ∈ P 1 ( I ) , ∫ ( u − Π h u ) (1) · v (1) ds = 0 . I 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. .. . .. . . .. . . .. . . .. . . . .. . .. . . .. . . .. . Error constant in Rayleigh quotient form . . 3. Eigenvalue problem of Bi-harmonic operator Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . .. . . . .. . . .. . . .. . . .. . . .. . . .. . . .. . .. . . . .. . . .. 37 / 79 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. .. .. . . .. . . . . . .. . .. .. . . .. .. . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . Definition of operators: (2D) . . Eigenvalue problem for 4th order differential operator . . Buckling eigenvalue problem Error constant in Rayleigh quotient form . .. . . . . . . . . . . .. . . .. . . .. . . .. .. . .. .. . . .. . .. . 38 / 79 . . .. . . .. . . Assumption: Ω is a simply connected bounded domain. ∆∆ u = − λ ∆ u, u = ∂u / ∂n = 0 on ∂ Ω , ∇ u = ( u x , u y ) for u being a scalar function. D 2 u = ( u xx , u xy , u yx , u yy ) for u being a scalar function. div p = p 1 ,x + p 2 ,y for p = ( p 1 , p 2 ) being a vector function. ∆ u = div · ∇ u = u xx + u yy curl p = p 2 ,x − p 1 ,y for p = ( p 1 , p 2 ) . curl u = ( u y , − u x ) for u being a scalar function. 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. .. .. . . .. . . . . . .. . . .. . . .. .. . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . Variational formulation: . . Eigenvalue problem for 4th order differential operator . . Buckling eigenvalue problem Error constant in Rayleigh quotient form . .. . . .. . . . . .. . . .. . . .. . . .. . . .. . . .. .. . . .. . .. . 38 / 79 . . .. . . .. . . Assumption: Ω is a simply connected bounded domain. ∆∆ u = − λ ∆ u, u = ∂u / ∂n = 0 on ∂ Ω , V := { v ∈ H 2 (Ω) | u = ∂u / ∂n = 0 on ∂ Ω } . ( D 2 u, D 2 v ) Ω = λ ( ∇ u, ∇ v ) Ω Find u ∈ V and λ ≥ 0 s.t. ∀ v ∈ V 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . . .. . . .. . . .. . . .. . . .. . .. .. . . .. . . .. . Error constant in Rayleigh quotient form . . . 2) Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . .. . .. . . .. . . .. . . . . . .. . . .. . . .. . . . .. . 39 / 79 .. . . .. . .. . Eigenvalue problem in FEM spaces V h Fujino-Morley FEM space: V h ( ̸⊂ V ) The function v h of V h satisfies, 1) v h | K ∈ P 2 ( K ) on each element K ∈ T h ; ∂v h ∂v h ∫ ∫ ∂n ds is continuous on interior edges; ∂n ds = 0 on boundary edges; e e 3) v h is continuous on interior nodes;. v h = 0 on boundary nodes. 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . .. . . .. . . .. . . .. . . .. . .. . . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . 2) . . Error constant in Rayleigh quotient form . . .. . . .. . . .. .. .. . . .. . . . . . .. . . .. . . .. . . . .. . . . .. . 39 / 79 .. . . .. .. . . Eigenvalue problem in FEM spaces V h Fujino-Morley FEM space: V h ( ̸⊂ V ) The function v h of V h satisfies, 1) v h | K ∈ P 2 ( K ) on each element K ∈ T h ; ∂v h ∂v h ∫ ∫ ∂n ds is continuous on interior edges; ∂n ds = 0 on boundary edges; e e 3) v h is continuous on interior nodes;. v h = 0 on boundary nodes. Define bilinear forms M ( · , · ) and N ( · , · ) over V h : ∑ ( D 2 u h , D 2 v h ) K , ∑ M ( u h , v h ) = N ( u h , v h ) = ( ∇ u h , ∇ v h ) K . K ∈T h K ∈T h For u, v ∈ H 2 (Ω) , M ( u, v ) = ( D 2 u, D 2 v ) , N ( u, v ) = ( Du, Dv ) . 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. .. . . .. . . .. .. . .. . . .. . . . . . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . . 2) . . . Error constant in Rayleigh quotient form . .. . . .. .. . . . . .. . . .. .. . .. .. . . .. . . . 39 / 79 . . . .. . .. . . .. . . . .. . . .. . Eigenvalue problem in FEM spaces V h Fujino-Morley FEM space: V h ( ̸⊂ V ) The function v h of V h satisfies, 1) v h | K ∈ P 2 ( K ) on each element K ∈ T h ; ∂v h ∂v h ∫ ∫ ∂n ds is continuous on interior edges; ∂n ds = 0 on boundary edges; e e 3) v h is continuous on interior nodes;. v h = 0 on boundary nodes. Eigenvalue problem in V h Find u h ∈ V h and λ h ≥ 0 s.t. ∀ v h ∈ V h . M ( u h , v h ) = λ h N ( u h , v h ) 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . . . .. . . .. . . .. . . .. . .. . . . .. . . .. . . .. . Error constant in Rayleigh quotient form . . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan .. .. . . . . .. . .. .. . . .. . . .. . . .. .. . . . .. . . 40 / 79 . . .. . . .. . The Fujino-Morley interpolation Π h On triangle element K , (Π h u ) | K is a quadratic function such that ∫ ( u − Π h u )( p i ) = 0 , ∇ ( u − Π h u ) · n ds = 0 ( i = 1 , 2 , 3) . e i ♣ ✶ ❑ ❡ ❡ ✸ ✷ ♣ ❡ ♣ ✷ ✶ ✸ 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . .. . . .. . . .. . . .. . . .. . .. . . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan Thus, . . . . Error constant in Rayleigh quotient form . . .. . . .. . . .. .. .. . . .. . . .. . . .. . . .. . . . . . . .. . . .. . . 40 / 79 .. .. . . . .. . The Fujino-Morley interpolation Π h On triangle element K , (Π h u ) | K is a quadratic function such that ∫ ( u − Π h u )( p i ) = 0 , ∇ ( u − Π h u ) · n ds = 0 ( i = 1 , 2 , 3) . e i Important property of Π h For u ∈ H 2 (Ω) , ∀ v h ∈ V h . ( D 2 ( u − Π h u ) , D 2 v h ) K = 0 , 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . .. . . .. .. . .. . . .. . . .. . .. . . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan Thus, . . . . Error constant in Rayleigh quotient form . . .. . . .. . . . .. .. . . .. . . .. . . . . . .. . . .. . . . .. . . .. . . 40 / 79 .. . .. . . . .. The Fujino-Morley interpolation Π h On triangle element K , (Π h u ) | K is a quadratic function such that ∫ ( u − Π h u )( p i ) = 0 , ∇ ( u − Π h u ) · n ds = 0 ( i = 1 , 2 , 3) . e i Important property of Π h For u ∈ H 2 (Ω) , ∀ v h ∈ V h . ∑ ( D 2 ( u − Π h u ) , D 2 v h ) K = 0 , K ∈T h 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . .. . . .. . .. .. . . .. . . .. . .. . . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan Thus, . . . . Error constant in Rayleigh quotient form . . .. . . .. . . . .. .. . . .. . . .. . . .. . . .. . . . . . . .. . . .. . . 40 / 79 . . .. . .. .. . The Fujino-Morley interpolation Π h On triangle element K , (Π h u ) | K is a quadratic function such that ∫ ( u − Π h u )( p i ) = 0 , ∇ ( u − Π h u ) · n ds = 0 ( i = 1 , 2 , 3) . e i Important property of Π h For u ∈ H 2 (Ω) , ∀ v h ∈ V h . ∑ ( D 2 ( u − Π h u ) , D 2 v h ) K = 0 , K ∈T h ∀ v h ∈ V h . M ( u − Π h u, v h ) = 0 , 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . . . .. . . .. . . .. . . .. . . .. .. . . , define Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan sup : over Constant by of .. Over . . Error constant . Error constant in Rayleigh quotient form . . .. .. .. . . . .. . . . . . .. . . .. . . . .. .. . . .. . . .. . . .. . . .. . . .. . 41 / 79 Interpolation error estimation for Π h On each element K , we consider the error estimation by using C 1 ( K ) : for u ∈ H 2 ( K ) . | u − Π h u | 1 ,K ≤ C 1 ( K ) | u − Π h u | 2 ,K where | · | d,K denotes the semi-norm of Sobolev function space H d ( K ) . 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . . .. . . .. . .. .. . . .. . . .. . .. .. . . .. . . .. . Error constant in Rayleigh quotient form . . . sup Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . . .. . . .. . . .. . . .. . . .. . . . . . .. . . . . .. .. .. .. . . . 41 / 79 . Interpolation error estimation for Π h On each element K , we consider the error estimation by using C 1 ( K ) : for u ∈ H 2 ( K ) . | u − Π h u | 1 ,K ≤ C 1 ( K ) | u − Π h u | 2 ,K where | · | d,K denotes the semi-norm of Sobolev function space H d ( K ) . Error constant C 1 ( K ) Over K of T h , define W ( K ) by ∫ ∂u W ( K ) := { u ∈ H 2 ( K ) | u ( p i ) = 0 , ∂nds = 0 , i = 1 , 2 , 3 } . e i Constant C 1 ( K ) over W ( K ) : | u | 1 ,K C 1 ( K ) := | u | 2 ,K , u ∈ W ( K ) 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . .. . . .. . . .. .. Error constant in Rayleigh quotient form .. . . .. . . .. . Notice . Notice that Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . Thus the unit norm vector along is the unit tangent vector and where , , we have for each . sup . : , About . max sup . . .. . . . .. . . .. . .. .. . . .. . . .. . . . 42 / 79 .. .. . . .. . .. . . . . . . . . .. .. . . Babuska-Aziz’s technique for upper bound of C 1 ( K ) ∥ u x ∥ 2 0 + ∥ u y ∥ 2 ( ∥ u x ∥ 2 , ∥ u y ∥ 2 ) C 1 ( K ) 2 = 0 0 0 ≤ | u x | 2 1 + | u y | 2 | u x | 2 | u y | 2 u ∈ W ( K ) u ∈ W ( K ) 1 1 1 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. .. . . .. . . .. . . .. . . .. . . .. . sup Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . . max sup Notice . Error constant in Rayleigh quotient form . .. . . .. .. . . . . .. . . .. . .. .. .. . . .. . . . 42 / 79 . . . . . .. . . . .. . . .. . . .. .. Babuska-Aziz’s technique for upper bound of C 1 ( K ) ∥ u x ∥ 2 0 + ∥ u y ∥ 2 ( ∥ u x ∥ 2 , ∥ u y ∥ 2 ) C 1 ( K ) 2 = 0 0 0 ≤ | u x | 2 1 + | u y | 2 | u x | 2 | u y | 2 u ∈ W ( K ) u ∈ W ( K ) 1 1 1 About u x , u y : Notice that u ( p i ) = 0 , we have for each i = 1 , 2 , 3 , ∫ ∫ ( u x , u y ) · ndx = ( u x , u y ) · τdx = 0 , e i e i where τ is the unit tangent vector and n the unit norm vector along e i . Thus ∫ ∫ u x ds = u y ds = 0 ( i = 1 , 2 , 3) e i e i 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. .. .. . . .. . . . . . .. . . .. . . .. .. max Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan Then: where sup . . sup . sup Notice Error constant in Rayleigh quotient form . .. . . .. . . . . .. . . .. . . . .. . . .. . . .. 42 / 79 . . . .. .. .. . . . . .. . . . .. .. . Babuska-Aziz’s technique for upper bound of C 1 ( K ) ∥ u x ∥ 2 0 + ∥ u y ∥ 2 ( ∥ u x ∥ 2 , ∥ u y ∥ 2 ) C 1 ( K ) 2 = 0 0 0 ≤ | u x | 2 1 + | u y | 2 | u x | 2 | u y | 2 u ∈ W ( K ) u ∈ W ( K ) 1 1 1 Recall the definition of constant C e ( K ) : ∥ v ∥ 0 ,K C e ( K ) := | v | 1 ,K , v ∈ V e ( K ) ∫ V e ( K ) := { v ∈ H 1 ( K ) | vds = 0 , i = 1 , 2 , 3 } e i ∥ u x ∥ 0 ≤ C e ( K ) | u x | 1 , ∥ u y ∥ 0 ≤ C e ( K ) | u y | 1 . 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. .. .. . . .. . . . . . .. . . .. . . .. .. max Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan Then: where sup . . sup . sup Notice Error constant in Rayleigh quotient form . .. . . .. . . . . .. . . .. . . . .. . . .. . . .. 42 / 79 . . .. . . . . .. . .. .. . . .. . . .. Babuska-Aziz’s technique for upper bound of C 1 ( K ) ∥ u x ∥ 2 0 + ∥ u y ∥ 2 ( ∥ u x ∥ 2 , ∥ u y ∥ 2 ) C 1 ( K ) 2 = 0 0 0 ≤ | u x | 2 1 + | u y | 2 | u x | 2 | u y | 2 u ∈ W ( K ) u ∈ W ( K ) 1 1 1 Recall the definition of constant C e ( K ) : ∥ v ∥ 0 ,K C e ( K ) := | v | 1 ,K , v ∈ V e ( K ) ∫ V e ( K ) := { v ∈ H 1 ( K ) | vds = 0 , i = 1 , 2 , 3 } e i ∥ u x ∥ 0 ≤ C e ( K ) | u x | 1 , ∥ u y ∥ 0 ≤ C e ( K ) | u y | 1 . C 1 ( K ) ≤ C e ( K ) 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . .. . .. . . .. . . .. . . .. . .. . . . .. . . .. . . .. . Error constant in Rayleigh quotient form Approximate computation shows that, Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan .. . . . . . .. . . .. . . .. . . .. . .. .. .. . . .. . . . . . .. . . .. . 43 / 79 Babuska-Aziz’s technique for upper bound of C 1 ( K ) For unit isoceles right triangle ˆ K , an easy-to-obtain lower bound: K ) ≤ 1 C e ( ˆ π . 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . . . .. . . .. . . .. . . .. . .. . . . .. . . .. . . .. . Error constant in Rayleigh quotient form Approximate computation shows that, Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan .. .. . . . . .. . . .. . . .. .. . .. . . .. .. . . . .. . . 43 / 79 . . .. . . . .. Babuska-Aziz’s technique for upper bound of C 1 ( K ) For unit isoceles right triangle ˆ K , an easy-to-obtain lower bound: K ) ≤ 1 C e ( ˆ π . K ) ≈ 0 . 2377 < 1 C 1 ( ˆ K ) ≈ 0 . 2338 < C e ( ˆ π ≈ 0 . 3183 . 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . Error constant in Rayleigh quotient form Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . .. .. . . . .. . . . . . .. . . .. . .. .. .. . . .. . . 44 / 79 . .. . . . .. . Contour lines of the constant C 1 ( K ) Dependence of constant C on triangle geometric parameters 1.0 0.8 a,b 0.6 0.4 0.2 (0 , 0) (1 , 0) 0.0 −1.0 −0.5 0.0 0.5 1.0 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . . .. . . .. . . .. .. . .. . . .. . .. .. . . .. . . .. . Error constant in Rayleigh quotient form . . . Hence, Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . . . . . .. . . . .. . . .. . . .. . 45 / 79 .. . . .. . . .. . . .. . . .. .. . . Error estimation for projection Π h Upper bound for C 1 ( K ) For general triangle K with the maximum inner angle as θ and the second longest edge length as L , C 1 ( K ) ≤ L √ 1 + | cos θ | π | u − Π h u | 1 ,K ≤ L √ 1 + | cos θ | | u − Π h u | 2 ,K π B { T αL θ A { O L 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . .. . . .. . .. .. . . .. . . .. . .. . . . .. . . .. . Error constant in Rayleigh quotient form Lower eigenvalue bounds based on Theorem 1 . Setting for application of Theorem 1 . . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . .. .. . . .. . . .. . . .. . . .. . . . . . . . .. . . .. 46 / 79 . .. . . .. . . .. V = H 2 0 (Ω) ; V h : Fujino-Morley FEM space ( V h ̸⊂ V ); K ∈T h ( D 2 u, D 2 v ) K ; M ( u, v ) := ∑ N ( u, v ) := ∑ K ∈T h ( ∇ u, ∇ v ) K ; Projection P h := Π h : M ( u − P h u, v h ) = 0 for v h ∈ V h . Error estimation for P h : ( ) | u − P h u | N ≤ C h | u − P h u | M C h := max K ∈T h C 1 ( K ) 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. .. .. . . .. . . . . . .. . .. .. . . .. .. . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . . From Theorem 1, we have, . Lower eigenvalue bounds . . Lower eigenvalue bounds based on Theorem 1 Error constant in Rayleigh quotient form . .. . . . . . . . . . . .. . . .. . . .. . . .. .. . .. .. . . .. .. . . 47 / 79 . . .. . . . . .. λ h,k ≤ λ k ( k = 1 , 2 , · · · , n ) . 1 + λ h,k C 2 h Recall the eigenvalue problem in V h Find u h ∈ V h and λ h ≥ 0 , s.t., ∀ v h ∈ V h . M ( u h , v h ) = λ h N ( u h , v h ) Eigenvalues: λ h, 1 ≤ λ h, 2 · · · ≤ λ h,n . 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. .. . .. . . .. . . .. . . .. . . . .. . .. . . .. . . .. . Error constant in Rayleigh quotient form . . General Fujino-Morley interpolation operators Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . .. . . . .. . . .. . . .. . . .. . . .. . . .. . . .. . .. . . . .. . . .. 48 / 79 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. .. .. . . .. . . . . . .. .. . .. . . .. .. . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan , For any . . Property 2) . 1) Extension of Fujino-Morley interpolation in 3D Fujino-Morley interpolation in 3D Error constant in Rayleigh quotient form . .. . . . . . . . .. . . .. . . .. . . .. . . .. . . .. .. . . .. . . .. . 49 / 79 .. . . . .. . . Let K be a tetrahedron with surfaces S i , i = 1 , 2 , 3 , 4 , and edges e i , i = 1 , · · · , 6 . For u ∈ H 2 ( K ) , Π h u ∈ P 2 ( K ) is determined by ∂ ( u − Π h u ) ∫ dS = 0 , i = 1 , 2 , 3 . S i ∂n ∫ e i u − Π h uds = 0 , i = 1 , · · · , 6 . 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . . . .. . . .. . . .. . . .. . . .. . . 1) Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . Property . 2) Extension of Fujino-Morley interpolation in 3D .. Fujino-Morley interpolation in 3D Error constant in Rayleigh quotient form . .. . . .. .. . . . . .. . . .. . . .. . . .. . . .. .. . . . . .. . . .. .. 49 / 79 . . . .. .. . . Let K be a tetrahedron with surfaces S i , i = 1 , 2 , 3 , 4 , and edges e i , i = 1 , · · · , 6 . For u ∈ H 2 ( K ) , Π h u ∈ P 2 ( K ) is determined by ∂ ( u − Π h u ) ∫ dS = 0 , i = 1 , 2 , 3 . S i ∂n ∫ e i u − Π h uds = 0 , i = 1 , · · · , 6 . For any v ∈ P 2 ( K ) , ∫ D 2 ( u − Π h u ) · D 2 vdX = 0 K 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . . .. . . .. . .. .. . . .. . . .. . . . . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan , For any . . Property . . . Extension of Fujino-Morley interpolation in 1D . Fujino-Morley interpolation in 1D Error constant in Rayleigh quotient form . .. .. . .. .. . . . .. . . . . . .. . . .. . . . .. .. . . .. . . .. . . 50 / 79 .. . . .. . . .. . Let I be an interval with two vertices a and b . For u ∈ H 2 ( I ) , Π h u ∈ P 2 ( I ) is determined by 1) (Π h u ) (1) ( a ) = u (1) ( a ) , (Π h u ) (1) ( b ) = u (1) ( b ) . 2) (Π h u )( a ) = u ( a ) , (Π h u )( b ) = u ( b ) . 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. .. .. . . .. . . . . . .. . . .. . . .. .. . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . Property . . Extension of Fujino-Morley interpolation in 1D . . Fujino-Morley interpolation in 1D Error constant in Rayleigh quotient form . .. . . .. . . . . .. . . .. . . .. . . .. . . .. . . .. .. . . .. . .. . 50 / 79 . . . . . .. . .. Let I be an interval with two vertices a and b . For u ∈ H 2 ( I ) , Π h u ∈ P 2 ( I ) is determined by 1) (Π h u ) (1) ( a ) = u (1) ( a ) , (Π h u ) (1) ( b ) = u (1) ( b ) . 2) (Π h u )( a ) = u ( a ) , (Π h u )( b ) = u ( b ) . For any v ∈ P 2 ( I ) , ∫ ( u − Π h u ) (2) · v (2) ds = 0 I 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. .. . .. . . .. . . .. . . .. . . . .. . .. . . .. . . .. . Error constant in Rayleigh quotient form . . Upper bound for eigenvalues of Bi-harmonic operator Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . .. . . . .. . . .. . . .. . . .. . . .. . . .. . . .. . .. . . . .. . . .. 51 / 79 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. .. .. . . .. . . . . . .. . .. .. . . .. .. . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . Variational formulation: II . . Variational formulation: I . . Upper bound for eigenvalues of Bi-harmonic operator Error constant in Rayleigh quotient form . .. . . . . . . . . . . .. . . .. . . .. . . .. .. . .. .. . . .. . .. . 52 / 79 . . . . .. . . .. V := { v ∈ H 2 (Ω) | u = ∂u / ∂n = 0 on ∂ Ω } . ( D 2 u, D 2 v ) = λ ( ∇ u, ∇ v ) Find u ∈ V and λ ≥ 0 s.t. ∀ v ∈ V D := { ( p 1 , p 2 ) ∈ H 1 0 (Ω) 2 | curl p = 0 in Ω } . Find u ∈ D and λ ≥ 0 s.t. ( ∇ p, ∇ q ) = λ ( p, q ) ∀ q ∈ D. 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . Variational formulation in FEM space Error constant in Rayleigh quotient form . .. . . .. . . .. . . .. .. . .. . . . .. ) in an Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan -conforming FEMs. The 4th order differential problem is solved by using . is given as , an upper bound for Since increasing order. ( . The eigenvalues of the above problem are denoted by s.t. and Find . . : Eigenvalue problem in . . . . .. . . .. . . .. . .. . . . .. . . .. . . . 53 / 79 . .. .. . . .. . . .. . . .. . . . . .. . . .. Lagrange FEM space over triangulation T h h = { v h ∈ H 1 (Ω) | v h | K ∈ P d ( K ) for K ∈ T h } L d ( L d h × L d ) D h = ∩ D h 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. .. .. . . .. . . .. . . .. . . .. . . . .. . increasing order. Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan -conforming FEMs. The 4th order differential problem is solved by using . is given as , an upper bound for Since . . . . . . . Variational formulation in FEM space Error constant in Rayleigh quotient form .. . . .. .. . . .. .. . . . . . .. . . .. . . . 53 / 79 .. . . . .. . . .. . . .. . . . .. . . .. Lagrange FEM space over triangulation T h h = { v h ∈ H 1 (Ω) | v h | K ∈ P d ( K ) for K ∈ T h } L d ( L d h × L d ) D h = ∩ D h Eigenvalue problem in D h : Find p h ∈ D h and ˆ λ h ≥ 0 s.t. ( ∇ p h , ∇ q h ) = ˆ λ h ( p h , q h ) ∀ q h ∈ D h . The eigenvalues of the above problem are denoted by ˆ λ h,k ( k = 1 , · · · , n ) in an 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . . . .. . .. .. . . .. . . .. . . .. . . . Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan increasing order. . . . . . .. Variational formulation in FEM space Error constant in Rayleigh quotient form . .. . . .. . . . . . .. . . .. . . .. . . .. . . .. .. . . . . .. . . .. .. 53 / 79 . . .. . . . .. Lagrange FEM space over triangulation T h h = { v h ∈ H 1 (Ω) | v h | K ∈ P d ( K ) for K ∈ T h } L d ( L d h × L d ) D h = ∩ D h Eigenvalue problem in D h : Find p h ∈ D h and ˆ λ h ≥ 0 s.t. ( ∇ p h , ∇ q h ) = ˆ λ h ( p h , q h ) ∀ q h ∈ D h . The eigenvalues of the above problem are denoted by ˆ λ h,k ( k = 1 , · · · , n ) in an Since D h ⊂ D , an upper bound for λ k is given as ˆ λ h,k . The 4th order differential problem is solved by using H 1 0 -conforming FEMs. 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . Computation results Error constant in Rayleigh quotient form . .. . . .. . Table : .. . . .. . . .. . Domain: a unit isosceles right triangle domain. Rough eigenvalue bounds .. [139.3179, – ] Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . Parameters in computing . [247.0735, – ] [204.8289, – ] 1/64 h [244.7311, 250.1080 ] [202.6644, 207.5471] [138.5584, 140.3105 ] 1/32 [235.5488, 257.0632 ] [194.1698, 213.8170] [135.6175, 142.5736 ] 1/16 . .. . . . . . .. . . .. . .. . . . .. . . .. . . .. 54 / 79 . .. .. . . .. . . .. .. . . . . .. . . .. . . λ 1 λ 2 λ 3 Upper bound computation: D h = ( L 2 h × L 2 h ) ∩ D , i.e., d = 2 . Constant C h for P h : C h := 0 . 24 h . 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. .. . .. . . .. . . .. . . .. . . . .. . .. . . .. . . .. . Error constant in Rayleigh quotient form . . Another eigenvalue problem for Bi-harmonic operators Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . .. . . . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. 55 / 79 ∆ 2 u = λu in Ω ; u = ∂u / ∂n = 0 on ∂ Ω . 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . .. . . .. . . .. . . .. . . .. . .. . . . .. . . .. . Error constant in Rayleigh quotient form . Eigenvalue problem of Bi-harmonic operator . . Variational formulation: Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . .. .. .. . . .. . . .. . . .. . . .. . . . . . . .. . . . .. 56 / 79 . . . . .. . .. .. Over bounded domain Ω ( ⊂ R 2 ), u = ∂u ∆ 2 u = λu in Ω; ∂n = 0 on ∂ Ω V := { v ∈ H 2 (Ω) | u = ∂u / ∂n = 0 on ∂ Ω } . ( D 2 u, D 2 v ) Ω = λ ( u, v ) Ω Find u ∈ V and λ ≥ 0 s.t. ∀ v ∈ V Let Π h be Fujino-Morley interpolation operator. We need to give estimation for constant C h such that ∥ u − Π h u ∥ 0 , Ω ≤ C h | u − Π h u | 2 , Ω . 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . .. . . .. .. . .. . . .. . . .. . .. . The error estimation for Fujino-Morley interpolation Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan where sup . . . Error constant in Rayleigh quotient form . . .. . . .. . . . .. .. . . .. . . .. . . . . . .. . . .. . . . .. . . . .. 57 / 79 . .. . . .. .. . . operator Π h Error constant C 0 ( K ) : ∥ u − Π h u ∥ 0 ,K ≤ C 0 ( K ) | u − Π h u | 2 ,K Constant C 0 ( K ) is characterized by | u | 0 ,K C 0 ( K ) := | u | 2 ,K , u ∈ W ( K ) ∫ ∂u W ( K ) := { u ∈ H 2 ( K ) | u ( p i ) = 0 , ∂nds = 0 , i = 1 , 2 , 3 } . e i 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . .. . . .. . . .. .. . .. . . .. . .. . . . .. . . .. . Error constant in Rayleigh quotient form Thus sup sup By using Taylor’s expansion, we can easily show that for unit isosceles right triangle, Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . .. . . .. . . .. . . .. .. . . .. . . . 58 / 79 . .. .. . . .. . . .. . . .. . . . . Upper bound of C 0 ( K ) Introduce an auxiliary space ˜ W ( K ) ⊂ W ( K ) : ˜ W ( K ) := { u ∈ H 2 ( K ) | u ( p i ) = 0 , i = 1 , 2 , 3 } . | u | 0 ,K | u | 0 ,K C 0 ( K ) := ≤ . | u | 2 ,K | u | 2 ,K u ∈ W ( K ) u ∈ ˜ W ( K ) √ 2 C 0 ( K ) ≤ ≤ 0 . 71 . 2 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . .. . . .. . . .. .. . .. . . .. . .. . . . .. . . .. . Error constant in Rayleigh quotient form Thus sup sup By using Taylor’s expansion, we can easily show that for unit isosceles right triangle, Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . .. . . .. . . .. . . .. .. . . .. . . . 58 / 79 . .. .. . . .. . . .. . . .. . . . . Upper bound of C 0 ( K ) Introduce an auxiliary space ˜ W ( K ) ⊂ W ( K ) : ˜ W ( K ) := { u ∈ H 2 ( K ) | u ( p i ) = 0 , i = 1 , 2 , 3 } . | u | 0 ,K | u | 0 ,K C 0 ( K ) := ≤ . | u | 2 ,K | u | 2 ,K u ∈ W ( K ) u ∈ ˜ W ( K ) √ 2 0 . 0909 ≈ C 0 ( K ) ≤ ≤ 0 . 71 . 2 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems

. . . .. .. . .. . . .. . . .. . . .. . .. .. . . .. . . .. . Error constant in Rayleigh quotient form Eigenvalue problem of plate Computation result: Over a unit isosceles right triangle domain Upper bound: conforming FEM space is needed. Liu Xuefeng Research Institue for Science and Engineering, Waseda Univeristy, Japan . . . .. . . .. . . .. . . .. . . .. . . . . . .. . . .. . . 59 / 79 .. . .. . . .. . Let λ h,k be the approximate eigenvalues from Fujino-Morely FEM. Lower bounds for eigenvalue λ k , λ h,k ≤ λ k . 1 + C 2 h λ h,k λ 1 ≥ 869 . 46 , λ 2 ≥ 2444 . 6 ( h = 1/64) . 自己共役楕円型偏微分作用素の高精度な固有値評価のフレームワーク An uniform approach to give high-precision eigenvalue estimation for self-adjoint eigenvalue problems