On angle conditions in the finite element method Michal Kˇ r´ ıˇ zek Institute of Mathematics, Academy of Sciences Prague, Czech Republic Joint work with Jan Brandts (University of Amsterdam), Antti Hannukainen (Aalto University, Finland), and Sergey Korotov (BCAM, Spain)
Abstract Angle conditions play important roles in the analysis of the fi- nite element method. They enable us to derive the optimal inter- polation order and prove convergence of the finite element method, to derive various a posteriori error estimates, to perform regular mesh refinements, etc. In 1968 Miloˇ s Zl´ amal introduced the min- imum angle condition for triangular elements. From that time a lot of other conditions appeared. We give a survey of various gen- eralizations of the minimum and also maximum angle condition in the finite element method.
M. Zl´ amal: On the Finite Element Method, Numer. Math. 12 ( 1968 ), 394–409 2 ∂ ∂u � � � − a ij + cu = f in Ω (polygon) ∂x i ∂x j i,j =1 u = 0 on ∂ Ω For quadratic triangular elements and smooth u Zl´ amal proved h 2 � u − u h � 1 ≤ C as h → 0 sin α 0 The minimum angle condition: α K ≥ α 0 > 0 ∀ K ∈ T h , where α K is the minimal angle of K .
− ∆ 2 u = f in Ω (polygon) u = ∂u ∂n = 0 on ∂ Ω For quintic triangular C 1 -elements and smooth u Zl´ amal proved h 4 � u − u h � 2 ≤ C as h → 0 sin 2 α 0 Thus the optimal rate of convergence is preserved if the same minimum angle condition holds.
Historical remarks The same condition was also introduced by ˇ Zen´ ıˇ sek in Appli- cations of Mathematics for the finite element method applied to a system of linear elasticity equations of second order published in 1969. However, this paper was submitted already on April 3, 1968, whereas Zl´ amal’s paper on April 17, 1968. In 1965 Kang Feng (in Chinese) proves convergence (not con- vergence rate) of the finite element method for triangular elements under the so-called maximum angle condition.
Martin Stynes in [Math. Comp. 1979, 1980] proved that the longest-edge bisection algorithm generates nonconforming trian- gulations satisfying the minimum angle condition. Longest-edge bisection algorithm may produce hanging nodes.
A generalization of the longest-edge bisection algorithm that generates conforming triangulations satisfying the minimum angle condition is given in [Hannukainen, Korotov, Kˇ r´ ıˇ zek, 2010]. This algorithm does not yield hanging nodes.
Various generalizations A generalization of the Zl´ amal minimum angle condition for tetrahedra is given in [Brandts, Korotov, Kˇ r´ ıˇ zek, 2008]: D α 1 F 2 α 5 α 6 A F 3 ϕ α 2 α 3 F 1 α 4 C B For simplicial elements in higher dimensions see [Brandts, Ko- rotov, Kˇ r´ ıˇ zek, 2011].
The so-called inscribed ball condition [Ciarlet, 1978] can be applied also to nonsimplicial elements in any dimension d : ∃ C > 0 ∀ T h ∈ F ∀ K ∈ T h : r K ≥ Ch K , where h K = diam K and r K is the radius of an inscribed ball b ⊂ K . In [J. Lin, Q. Lin, 2003], the inscribed ball condition was re- placed by a simpler condition on the volume of every element.
Equivalent conditions for simplicial elements for any d Condition 1: vol S ≥ C 1 h d ∃ C 1 > 0 ∀ T h ∈ F ∀ S ∈ T h : S . Condition 2: vol b ≥ C 2 h d ∃ C 2 > 0 ∀ T h ∈ F ∀ S ∈ T h : S , where b ⊂ S is the inscribed ball of S . Condition 3: ∃ C 3 > 0 ∀ T h ∈ F ∀ S ∈ T h : vol S ≥ C 3 vol B, where B ⊃ S is the circumscribed ball about S . See [Brandts, Křížek, 2003] and [Brandts, Korotov, Křížek, 2009, 2011].
Lower bound of the interpolation error Theorem [Kˇ r´ ıˇ zek, Roos, Chen, 2011]. If v ∈ C 3 (Ω) is not a linear polynomial and if the minimum angle condition holds, then there exists C > 0 such that | v − π h v | 1 ≥ Ch as h → 0 , where π h v is the standard linear interpolant of v .
Two-sided bounds of the discretization error zek, Roos, Chen, 2011]. If u ∈ C 3 (Ω) then for the Theorem [Kˇ r´ ıˇ piecewise linear FE-approximation of u over a family of uniform triangulations satisfying the minimum angle condition with h → 0 we get (1 − C 1 h ) | u − π h u | 1 ≤ | u − u h | 1 ≤ (1 + C 1 h ) | u − π h u | 1 , (1 − C 2 h ) � u − π h u � 1 ≤ � u − u h � 1 ≤ (1 + C 2 h ) � u − π h u � 1 . This can be generalized to bilinear elements [Q. Lin, J. Lin, 2006] and [Yan, 2008].
Maximum angle condition A cross section of an electrical motor has many planes of sym- metry.
In 1976 three groups [Babuˇ ska, Aziz], [Barnhill, Gregory], and [Jamet] independently for d = 2 discovered that γ K ≤ γ 0 < π ( maximum angle condition ) ⇒ � u − u h � 1 ≤ Ch | u | 2 i.e., the minimum angle condition is not optimal. By means of the Jensen, Minkowski, Cauchy-Schwarz and the triangle inequality we can derive that [Kˇ r´ ıˇ zek, 1991] Maximum angle condition ⇒ � u − u h � 1 ,p ≤ C p h | u | 2 ,p , p > 1 In [Synge, 1957] the following interpolation property is proved Maximum angle condition ⇒ � v − π h v � 1 , ∞ ≤ Ch | v | 2 , ∞
We need not be afraid of using triangles with small angles which can be useful for developing FE-software for adaptive mesh refine- ment, since we need not prescribe any lower positive bound upon the minimum angle. Thus we may employ triangular elements which are almost flat (degenerate). This can be useful for cover- ing thin slots, gaps or strips of different materials. “Flat” triangles are also commonly used to approximate func- tions that change much more rapidly in one direction than in another direction.
Example. Let v ( x, y ) = 2 x 2 − xy − 3 y 2 . h − 1 � v − π h v � 1 / | v | 2 j h � v − π h v � 1 0 1.414214 2.100264 0.203996 1 0.559017 0.732818 0.180067 2 0.257694 0.318461 0.169751 3 0.125973 0.153600 0.167485 4 0.062622 0.076344 0.167462
Definition. A family F = { T h } of face-to-face partitions into tetrahedra is said to satisfy the maximum angle condition if ∃ γ 0 < π ∀ T h ∈ F ∀ K ∈ T h : γ K ≤ γ 0 and ϕ K ≤ γ 0 , where γ K is the maximum angle of all triangular faces of K , and ϕ K is the maximum angle between faces of K . Optimal order of convergence is proved in [Kˇ r´ ıˇ zek, 1992].
a) b) The maximum angle conditions γ K ≤ γ 0 and ϕ K ≤ γ 0 are independent, see [Kˇ r´ ıˇ zek, 1992].
� � Almost degenerate tetrahedra can be used to cover thin slots or strips, which saves computer memory. They should also be used along polyhedron edges, where the true solution of the boundary value problem is usually “smooth” along edges and “nonsmooth” across edges. Thomas Apel: Anisotropic Finite Elements, Teubner, 1999. An elegant generalization of the maximum angle condition for d is presented in [Jamet, 1976]. Roughly speak- other elements in ing, we can still achieve the optimal approximation order if el- ements degenerate, except the case when all element edges are d . almost parallel to some hyperplane in
The maximum angle condition is not necessary for con- vergence of the finite element method x 2 A 3 A A 1 2 x 1 Take ε > 0 and the triangle K with vertices A 1 = ( − 1 , 0), A 2 = (1 , 0), and A 3 = (0 , ε ). Consider the function v ( x, y ) = x 2 and its linear interpolant π ε v on K . Then � ∂π ε v 0 = 1 ε 2 vol K = 1 2 � � � v − π ε v � 2 1 ≥ ε → ∞ as ε → 0 � � ∂y � See [Strang, Fix, 1973].
Spaces V h of linear elements over triangulations T h
For spaces W h ⊃ V h we have � u − u h � 1 ≤ w h ∈ W h � u − w h � 1 ≤ inf v h ∈ V h � u − v h � 1 inf ≤ C � u − π h u � 1 ≤ C ′ h | u | 2 where u h ∈ W h is the standard Galerkin approximation of the weak solution u ∈ H 2 (Ω), see [Hannukainen, Korotov, Kˇ r´ ıˇ zek, 2011].
On nonobtuse simplicial face-to-face partitions A single obtuse triangle in the partition can destroy the discrete maximum principle ( f ≥ 0 ⇒ u h ≥ 0) for the Poisson equation with zero boundary conditions (see [Brandts, Korotov, Kˇ r´ ıˇ zek, ˇ Solc, 2009]). ● ● ● The discrete maximum principle is of interest to avoid nega- tive numerical values of typical positive physical quantities like concentration, temperature (in Kelvins), density, and pressure.
B i F j α i j F i B j For linear simplicial elements on K we have ( ∇ v i ) ⊤ ∇ v j = − vol F i vol F j cos α ij , if i � = j ( d vol K ) 2 where v i are Courant basis functions (see [Kˇ r´ ıˇ zek, Lin, 1995], [Xu, Zikatanov, 1999], [Brandts, Korotov, Kˇ r´ ıˇ zek, 2007]). Nonobtuse simplicial partitions yield diagonally dominant stiffness matrices.
Acute simplicial partitions A given simplex is acute if all dihedral angles are less than 90 ◦ . A partition is said to be acute if all its simplices are acute. Theorem [Fiedler, 1957]. Let d > 2 . If a d -simplex is acute, then each of its facets is an acute ( d − 1) -simplex.
✁ ✁ ✁ Theorem [¨ 3 . Ung¨ or, 2001]. There exists an acute partition of Theorem [Kopczy´ nski, Pak, Przytycki, 2009]. There exists an acute partition of any Platonic solid. d Theorem [Kˇ r´ ıˇ zek, 2006, 2010]. There is no acute partition of for d ≥ 5 . 4 . Conjecture. There is no acute partition of
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