Error estimates for anisotropic finite elements and applications Ricardo G. Dur´ an Departamento de Matem´ atica Facultad de Ciencias Exactas y Naturales Universidad de Buenos Aires Argentina ICM, Madrid August 26, 2006 http://mate.dm.uba.ar/ ∼ rduran/
COWORKERS • Gabriel Acosta, UNGS and UBA, Argentina • Ariel L. Lombardi, UBA and CONICET, Argentina 1
OUTLINE OF THE TALK • Introduction to FEM • Basic error analysis and examples • The regularity hypothesis on the elements • Necessity of relaxing the regularity hypothesis • Error estimates for the Lagrange interpolation 2
• Differences between 2D and 3D cases • Necessity of other interpolations • An average interpolation • Results for mixed finite element and non-conforming meth- ods • Application to the Stokes equations • Application to problems with boundary layers 3
FINITE ELEMENT METHOD GENERAL SETTING: V Hilbert space B ( u, v ) = F ( v ) ∀ v ∈ V B continuous bilinear form, F continuous linear form. APPROXIMATE SOLUTION: V h finite dimensional space , u h ∈ V h B ( u h , v ) = F ( v ) ∀ v ∈ V h 4
ERROR ESTIMATES IN FINITE ELEMENT APPROXIMATIONS They can be divided in two classes • A PRIORI ESTIMATES • A POSTERIORI ESTIMATES 5
GOALS OF A PRIORI ESTIMATES • To prove convergence and to know the order of the error • To know the dependence of the error on different things (geometry of the mesh, regularity of the solution, degree of the approximation) A typical a priori error estimate is of the form � u − u h � ≤ Ch α �| u �| where h is a mesh size parameter. 6
A BASIC QUESTION IS: WHAT KIND OF ELEMENTS ARE ALLOWED? or, in other words, HOW DOES THE ERROR DEPEND ON THE GEOMETRY OF THE ELEMENTS? The classic theory is based in the so-called “REGULARITY ASSUMPTION” 7
h T ρ T h T ≤ σ ρ T h T exterior diameter, ρ T interior diameter The constant in the error estimates depends on the regularity parameter σ 8
The advantages of the arguments based on this hypothesis are: • It allows for very general results on error estimates for ap- proximations of different kinds • It implies the so called inverse estimates which simplify many arguments See for example the books by Ciarlet and Brenner-Scott 9
HOWEVER, In many applications it is essential to remove the regularity hy- pothesis on the elements and to use ANISOTROPIC OR FLAT ELEMENTS 10
EXAMPLE 1: PROBLEMS WITH BOUNDARY LAYERS 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 11
EXAMPLE 2: CUSPIDAL DOMAINS 12
The constants in error estimates depend on: • CONSTANTS IN INTERPOLATION OR BEST APPROX- IMATION ERROR • STABILITY CONSTANTS • BOUNDS OF CONSISTENCY TERMS IN NON-CONFORMING METHODS In standard analysis the regularity hypothesis is used for all these steps 13
CASE 1: COERCIVE FORMS AND CONFORMING METHODS V h ⊂ V If B ( v, v ) ≥ α � v � 2 ∀ v ∈ V then � u − u h � ≤ C inf � u − v � v ∈ V h The computed approximate solution is, up to a constant, like the best approximation. 14
CLASSIC EXAMPLES Scalar second order elliptic equations: ∂x i ( a ij ∂u ∂ − � n Ω ⊂ R n ∂x j ) = f in i,j =1 u = 0 on ∂ Ω n γ | ξ | 2 ≤ a ij ξ i ξ j ≤ M | ξ | 2 ∀ ξ ∈ R n � ∀ x ∈ Ω i,j =1 V = H 1 0 (Ω) 15
The linear elasticity equations: � Ω ⊂ R n − µ ∆ u − ( λ + µ ) ∇ div u = f in u = 0 on ∂ Ω � B ( u , v ) = Ω { 2 µε i,j ( u ) ε i,j ( v ) + λ div u div v } dx where ε i,j ( v ) = 1 + ∂v j 2( ∂v i ) ∂x j ∂x i V = H 1 0 (Ω) n 16
CASE 2: NON COERCIVE FORMS SATISFYING AN INF-SUP CONDI- TION AND CONFORMING METHODS B ( u, v ) inf sup � u �� v � ≥ α > 0 u ∈ V h v ∈ V h In this case we also have � u − u h � ≤ C inf � u − v � v ∈ V h 17
CLASSIC EXAMPLES 1-Mixed formulation of second order elliptic problems Ω ⊂ R n � div( a ( x ) ∇ p ) = f in p = 0 on ∂ Ω u = − a ( x ) ∇ p in Ω div u = f p = 0 on ∂ Ω � � � Ω a ( x ) − 1 u · v + B (( u , p ) , ( v , q )) := Ω p div v + Ω q div u V = H (div , Ω) n × L 2 (Ω) 18
2-The Stokes equations Ω ⊂ R n − ν ∆ u + ∇ p = f in Ω ⊂ R n div u = 0 in u = 0 on ∂ Ω B (( u , p ) , ( v , q )) = F ( v ) � � � B (( u , p ) , ( v , q )) := Ω ∇ u : ∇ v − Ω p div v − Ω q div u 0 (Ω) n × L 2 V = H 1 0 (Ω) 19
CASE 3: STABLE FORMS BUT NON-CONFORMING METHODS V h �⊂ V STRANG’S LEMMA: � � | B h ( u, w ) − F ( w ) | inf � u − v � + sup � u − u h � ≤ C � w � v ∈ V h w ∈ V h 20
CLASSIC EXAMPLE Crouzeix-Raviart linear non-conforming method For the Poisson equation: � � B h ( u, v ) = K ∇ u · ∇ v K The arguments used in the original paper of CR use the regularity assumption on the elements. 21
MAIN TOOLS TO PROVE THE INF-SUP 1- Brezzi’s theory for mixed methods For example, for the Stokes problem u h ∈ U h p h ∈ Q h it is enough to prove � Ω p div v inf sup ≥ α > 0 � p �� v � p ∈ Q h v ∈ U h or equivalently, the existence of the Fortin operator 22
0 (Ω) n − Π h : H 1 → U h such that � Ω div ( u − Π h u ) q = 0 ∀ q ∈ Q h and � Π h u � H 1 0 ≤ C � u � H 1 0 Again, many of the arguments to obtain this result make use of the regularity of the elements. 23
LAGRANGE INTERPOLATION Consider the lowest order case: K triangle , P 1 interpolation or K quadrilateral , Q 1 isoparametric interpolation u I ( P i ) = u ( P i ) P i nodes 24
THE REGULARITY HYPOTHESIS CAN BE REPLACED BY WEAKER ASSUMPTIONS! IN THE CASE OF TRIANGLES IT CAN BE REPLACED BY THE “MAXIMUM ANGLE CONDITION” First results: Babuska-Aziz, Jamet (1976) Other references: Krizek, Al Shenk, Dobrowolski, Apel, Nicaise, Formaggia, Perotto, Acosta, Lombardi, Dur´ an, etc.. BAD GOOD 25
IDEA: WORK WITH AN APPROPRIATE REFERENCE FAM- ILY INSTEAD OF A FIXED REFERENCE ELEMENT F k k α h h F : ˜ T − → T B ∈ R n × n a ∈ R n F (˜ x ) = B ˜ x + a 26
F k k h h B ∈ R n × n a ∈ R n F (˜ x ) = B ˜ x + a 27
(p,q) F b d 2 d 1 a K (a,b,p,q) K B ∈ R n × n a ∈ R n F (˜ x ) = B ˜ x + a 28
THE P 1 CASE Let ˆ T be the triangle with vertices at (0 , 0), (0 , 1) and (1 , 0) e type inequality: if ˆ ℓ is an edge of ˆ Poincar´ T then � ℓ v = 0 = ⇒ � v � L 2 ( ˆ T ) ≤ C �∇ v � L 2 ( ˆ T ) ˆ It follows from: Standard Poincar´ e inequality: � T v = 0 = ⇒ � v � L 2 ( ˆ T ) ≤ C �∇ v � L 2 ( ˆ T ) ˆ and 29
Trace theorem: � v � L 2 (ˆ ℓ ) ≤ C � v � H 1 ( ˆ T ) Changing variables: ˜ x = h ˆ x and ˜ y = k ˆ y we have k h � � ∂v ∂v � � � � � � � ℓ v = 0 = ⇒ � v � L 2 ( ˜ T ) ≤ C h + k � � � � � � ∂x ∂y � L 2 ( ˜ � T ) � L 2 ( ˜ T ) � 30
but, if ℓ = { 0 ≤ x ≤ h, y = 0 } , we have ∂ � ∂x ( u − u I ) = 0 ℓ and then ∂ 2 u ∂ 2 u � � � � ∂ � � � � � � � � ∂x ( u − u I ) ≤ C h + k � � � � � � ∂x 2 � � � � ∂x∂y � L 2 ( ˜ T ) � � L 2 ( ˜ � L 2 ( ˜ T ) T ) � � THE CONSTANT C IS INDEPENDENT OF h and k ! 31
Now, for a general triangle T F k k α h h F : ˜ T − → T B ∈ R n × n a ∈ R n F (˜ x ) = B ˜ x + a C � B − 1 � ≤ � B � ≤ C sin α Then C sin α h T � D 2 u � L 2 ( T ) �∇ ( u − u I ) � L 2 ( T ) ≤ 32
THE CASE Q 1 ON PARALLELOGRAMS F k k h h As in the case of triangles we obtain ∂ 2 ( u − u I ) ∂ 2 ( u − u I ) � � � � ∂ � � � � � � � � ∂x ( u − u I ) ≤ C h + k � � � � � � ∂x 2 � � � � ∂x∂y � L 2 ( R ) � � L 2 ( R ) � L 2 ( R ) � � ∂ 2 u I ∂ 2 u I ∂x 2 = 0 but ∂x∂y � = 0 33
However, ∂ 2 u I ∂ 2 u � � ∂x∂y = ∂x∂y R R and so ∂ 2 u I ∂ 2 u � � � � � � � � ≤ � � � � � � � � ∂x∂y ∂x∂y � L 2 ( R ) � L 2 ( R ) � � REMARK: The fact that D 2 u I � = 0 introduces an extra difficulty. A similar difficulty arises in the analysis of mixed methods (and as we will see, that case is more complicated) 34
Then ∂ 2 u ∂ 2 u � � � � ∂ � � � � � � � � ∂x ( u − u I ) ≤ C h + k � � � � � � ∂x 2 � � � � ∂x∂y � L 2 ( R ) � � L 2 ( R ) � L 2 ( R ) � � and for a general parallelogram C sin α h T � D 2 u � L 2 ( P ) �∇ ( u − u I ) � L 2 ( P ) ≤ 35
THE CASE OF QUADRILATERALS IS MORE COMPLICATED SEVERAL CONDITIONS HAVE BEEN INTRODUCED - Ciarlet-Raviart (1972): Regularity and non degeneracy of the angles. - Jamet (1977): Regularity. - Zenizek-Vanmaele (1995), Apel (1998): Allows anisotropic (flat) elements but far from triangles. 36
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