Numerical methods for PDEs FEM convergence, error estimates, piecewise polynomials

Transcription

1 Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Numerical methods for PDEs FEM convergence, error estimates, piecewise polynomials Dr. Noemi Friedman

2 Contents of the course Fundamentals of functional analysis Abstract formulation FEM Spatial (meshing) and functional discretization (the basis functions) Convergence, regularity Variational crimes Implementation Mixed formulations (e.g. Stokes) Stabilisation for flow problems Error indicators/estimation Adaptivity FEM and its convergence Dr. Noemi Friedman PDE2 Seite 2

3 Abstract formulation, examples FEM and piecewise polynomials Non-degenerate triangulation, refinement of triangulation Convergence using piecewise linear functions Convergence using piecewise higher order elements Implementation, sparsity of the stiffnes matrix Variational crime: numerical integration Variational crime: curved boundaries FEM and its convergence Dr. Noemi Friedman PDE2 Seite 3

4 Recap: boundedness of the error of Galerkin method a u u h, v = 0 v V h If a, is an inner product, it means that the approximation is an orthogonal projection to the subspace in the energy norm: error = u x u h x E u x v x E v x V h According to Céa s theorem (see prove at the lecture note), even without a, being symmetric, the error of the approximation of Galerkin will be allways bounded: u u h M δ u v v V h Where M and δ are constants from the conditions of boundedness and V-ellipticity of the bilinear term a, : a u, v M u v a u, u δ u 2 FEM and its convergence Dr. Noemi Friedman PDE 2 Seite 4

5 Recap: what do we have to solve? [Chapter 4] N i=1 c i Ψ i (x) Ψ j x dω = Ω f x Ψ j x dω Ω Kc = f K ij f j FEM: Galerkin method where Ψ j are piecewise polynomials Main goals when implementing: efficient calculation of K efficient calculation of f solve Kc=f efficiently true solution is approximated well (error is small enough) piecewise polynomials FEM and its convergence Dr. Noemi Friedman PDE 2 Seite 5

6 Piecewise polynomials and the FEM [Chapter 4.1] piecewise polynomials: a function that is defined by a polynomial on each subdomain mesh: the collection of subdomains nodal basis: 1 i = j ψ i x j = 0 i j ψ i x j, y j = 1 i = j 0 i j ψ 2 ψ 3 ψ 4 ψ 5 u x u h x = 4 i=1 u i ψ i x u h (x) P h 1 continious piecewise linear functions for Poisson equation we have to show that P h 1 H 1 when it s not satisfied it is not a conforming element method FEM and its convergence Dr. Noemi Friedman PDE 2 Seite 6

7 Recap: Why linear piecewise polynomials defined on a [Chapter 4.1.1] triangulation are in the subspace Suppose, that uniqueness and existence can be shown in H 1 Show, that the space of all continious piecewise linear functions defined on the triangulation T h is a subspace of H 1. The continious piecewise linear functions can be writen in the general form: u = a i + b i x + c i y (x, y) T i a j + b j x + c j y = a i + b i x + c i y x, y e = T i T j, We have to show that all us are in H 1 : Ω u 2 < Ω = i T i Ω u x 2 + u y 2 < FEM and its convergence Dr. Noemi Friedman PDE 2 Seite 7

8 Recap: Why linear piecewise polynomials defined on a triangulation are in the subspace 1) First let s show that u 2 < Ω u 2 = Ω i T i a i + b i x + c i y 2 < i T i max a i + b i x + c i y 2 T i < 2.)Show that Ω u 2 < Ω = i T i There is no derivatives in a strong sense, but let s see whether these give weak derivatives: g 1 : = u x = b i (x, y) int(t i ) g 2 : = u y = c i (x, y) int(t i ) FEM and its convergence Dr. Noemi Friedman PDE 2 Seite 8

9 Recap: Why linear piecewise polynomials defined on a triangulation are in the subspace are weak derivatives if for any v C 0 Ω u v x = Ω vg 1 and Ω u v y = Ω vg 2 holds. We prove only this, because the approach of the derivation for g 2 is same. Let s first look at the left hand side of the equation: Ω u v x = i T i u v x = i T i uv n 1 + T i u x v uv n 1 = 0 on the edges of the boundary, and uv n 1 = T i T i On the common edges of neighboring triangles T i and T j T j uv n 1 i T i uv n 1 = 0 T i u x v = T i b i v FEM and its convergence Dr. Noemi Friedman PDE 2 Seite 9

10 Recap: Why linear piecewise polynomials defined on a triangulation are in the subspace The left hand side of the equation: Ω u v x = vg 1 Ω Ω u v x = i T i b i v The left hand side of the equation: Ω vg 1 = i T i b i v FEM and its convergence Dr. Noemi Friedman PDE 2 Seite 10

11 Piecewise polynomials and the FEM [Chapter 4.1] Derivatives in the classical sense: weak derivatives of v (see proof in Gockenbach Chapter 4.1) v x, y L 2 v x, y L 2 P h 1 H 1 Ψ 1, Ψ 19 0 K 1,19 = Ψ 1 x Ψ 19 x dω = 0 Ω FEM and its convergence Dr. Noemi Friedman PDE 2 Seite 11

12 Sparsity of the stiffness matrix [Chapter 4.1] K ij = Ψ i x Ψ j x dω 0 Ω if nodes i and j are adjacent Example: K 13,j 0 if j = 7,8,12,13,14,18,19 Similarly: K i,13 0 if i = 7,8,12,13,14,18,19 max 7 nonzero elements/row and /column FEM and its convergence Dr. Noemi Friedman PDE 2 Seite 12

13 Quadratic piecewise polynomials Ψ i x, y = Ψ 5 Ψ 18 [Chapter 4.2] 6 nodes needed FEM and its convergence Dr. Noemi Friedman PDE 2 Seite 13

14 Quadratic piecewise polynomials [Chapter 4.2] nz = = 16% nz = = 5% FEM and its convergence Dr. Noemi Friedman PDE 2 Seite 14

15 Cubic piecewise polynomials [Chapter 4.3] FEM and its convergence Dr. Noemi Friedman PDE 2 Seite 15

16 Higher order piecewise polynomials Requirements for polynomial of degree d in 2D (two variables) number of nodes per edge (to guarantee continuity of the ansatz function): d + 1 (d 1 in between the vertices) 3d on the edges Number of parameters needed to define 2D polynomials x 3 x 1 x 2 xy y 2 y x 2 y xy 2 y 3 Co Lin x 4 x 3 y x 2 y 2 xy 3 y 4 d = 4 d = 6 d = 5 FEM and its convergence Dr. Noemi Friedman PDE 2 Seite 16

17 Higher order piecewise polynomials Let s check setup for d = 4 in 2D (two variables) number of nodes per edge (to guarantee continuity of the ansatz function): d + 1 = 5 (d 1=3 in between the vertices) 3d = 12 on the edges Number of parameters needed to define 2D polynomials x 3 x 1 x 2 xy y 2 y x 2 y xy 2 y 3 Co Lin (d + 1)(d + 2) 2 = 15 Number of nodes in the middle x = 3 x 3 y x 2 y 2 xy 3 y 4 d = 4 FEM and its convergence Dr. Noemi Friedman PDE 2 Seite 17

18 Higher order piecewise polynomials The a general piecewise polynomial takes the form u = (i) a 1 + (i) (i) + a 2 x + a3 y + (i) + a 4 x 2 (i) (i) + a 5 xy + a6 y 2 + (i) + a 7 x 3 (i) + a 8 x 2 (i) y + a 9 xy 2 (i) + a 10y 3 + (i) + a 11 x 4 (i) + a 12 x 3 (i) y + a 13 x 2 y 2 (i) + a 14 xy 3 + a 14 (i) y 4 (x, y) T i 12 Let s suppose I know the solution at the nodes x j, y j j=1 : u j = u(x j, y j ) Then the nodes should define uniquely the plane, that is, the values for a k (i) 12 j=1 d = 4 FEM and its convergence Dr. Noemi Friedman PDE 2 Seite 18

19 Higher order piecewise polynomials The a general piecewise polynomial takes the form u 1 u 2 1 x 1 y 1 2 x 1 x 1 y 1 2 y 1 1 x 2 y 2 2 x 2 x 2 y 2 2 y 2 3 x 1 4 y 1 3 x 2 4 y 2 a 1 (i) a 2 (i) (x j, y j ) T i = u x 12 y 12 (i) a 12 W W is not singular d = 4 FEM and its convergence Dr. Noemi Friedman PDE 2 Seite 19

20 Different discretisations of the functional space Shape functions in C 0 and in C 1 on 1D and or on 2D (quadratic or triangular) elements, the Lagrange polynomials and the Hermite polynomials, number of basis functions, conforming elements (see lecture, or more in [1], [2], [3]) Picture source: [1]: Brenner&Scott: The Mathematical Theory of FEM, Chapter 3 Construction of the finite element space [2]: Zienkiewicz&Taylor: The Finite Elment Method, Chapter 8. Standard and hierarchical element shape functions: some general families of C_0 continuity [3]: Logg&Mardal&Wells: Automated Solution of Differential Equations by the FEM- The Fenics Book, Chapter 3: Common and unusual finite elements FEM and its convergence Dr. Noemi Friedman PDE 2 Seite 20

21 1D Example with linear nodal basis p(x) p x = ax l Strong form: l/5 l/5 l/5 l/5 l/5 u 0 = u l = 0 Weak form: Discretisation of the weak form: l EA du l dψ dx dx dx = p x ψ x dx 0 0 u x 4 i=1 u i ψ i (x) Not efficient to calculate all the elements of the stiffness matrix one by one! 4 i=1 u j EA l ψ i (x) x ψ j (x) dx = x l p(x)ψ j (x)dx Calculate element stiffness matrices and assemble K ij f j FEM and its convergence Dr. Noemi Friedman PDE 2 Seite 21

22 1D Example with linear nodal basis l p(x) 1 l/5 2 l/5 3 l/5 4 l/5 5 l/5 6 instead: Compute stiffness matrix elementwisely and then assemble Global stiffness matrix ψ 2 K 4 e 1,1 = EA K 4 e 1,2 = EA K 4 e 2,1 = EA K 4 e 2,2 = EA ψ 3 ψ 4 (x) Ω 4 x ψ 4 (x) Ω 4 x ψ 5 (x) Ω 4 x ψ 5 (x) Ω 4 x ψ 4 (x) dx x ψ 5 (x) dx x ψ 4 (x) dx x ψ 5 (x) dx x ψ 4 ψ K e 2 1,1 K e K e 2 (1,2) 1 (2,2) K e 3 1,1 K e K e K e 2 (2,1) 2 (2,2) 3 (1,2) K e K e 4 (1,1) 4 (1,2) K e 3 (2,1) K e 3 (2,2) K 4 e (2,1) K 4 e (2,2) K 5 e (1,1) u 1 u 2 u 3 u 4 u 5 = 0 f 2 f 3 f 4 f 5 K e 4 = 4 K 4 e (1,1) K 4 e (1,2) 6 1 u K 4 e (2,1) K 4 e (2,2) K u FEM and its convergence Dr. Noemi Friedman PDE 2 Seite 22 f

23 1D Example with linear nodal basis l p(x) instead: Compute stiffness matrix elementwisely and then assemble 1 l/5 2 l/5 3 l/5 4 l/5 5 l/ ψ 2 ψ 3 ψ 4 ψ u 1 = 0 f 4 e 1 = Ω 4 p(x)ψ 4 (x)dx 2 K e 2 1,1 K e K e 2 (1,2) 1 (2,2) u 2 f e 1 2 f e 2 1 f 4 e 2 = Ω 4 p(x)ψ 5 (x)dx 3 K e 3 1,1 K e 2 (2,1) K e 2 (2,2) K 3 e (1,2) u 3 f e 2 2 f e K e K e 4 (1,1) 4 (1,2) K e 3 (2,1) K e 3 (2,2) u 4 f 3 e 2 f 4 e 1 f e 4 = 4 f 4 e K 4 e (2,1) K 4 e (2,2) K 5 e (1,1) 1 u 5 u 6 f e 4 2 f e f 4 e 2 K u FEM and its convergence Dr. Noemi Friedman PDE 2 Seite 23 f

24 Non-degenerate triangulation [Chapter 5.1.2] Diameter of a set: Diameter of a triangle: D T : length of longest side d T : largest circle contained in T d T D T : measures how skinny the triangle is Other definitions Τ h :triangulation (set of triangles), with h: maximal diameter of any triangle in Τ h (the length of longest side) Nondegenerate triangulation: for all the triangles in the triangulation. FEM and its convergence Dr. Noemi Friedman PDE 2 Seite 24

25 Convergence using piecewise polynomials error = u u h E u v E v V h u u h M δ u v v V h Let s compare the best approximation u h with the proximodel with piecewise linear functions: u I n x = j=1 u Ij Ψ j x u I V h u u h E u u I E u u h M δ u u I If I can bound the expression in the r.h.s, I also bound the errors. FEM and its convergence Dr. Noemi Friedman PDE 2 Seite 25

26 Convergence using piecewise linear functions [Chapter 5.1] Theorem: {T h } : non-degenerate family of triangulations of a polygonal domain Ω R 2 u H 2 u I : piecewise linear approximation There exists a constant C depending on Ω and the value ρ (see definition of nondegenerate triangulation) such that u u I u u I L2 Ch 2 u H2 H1 Ch u H2 where: (seminorm) FEM and its convergence Dr. Noemi Friedman PDE 2 Seite 26

27 Convergence using piecewise linear functions [Chapter 5.1] Source: Gockenbach: Understanding and Implementing FEM FEM and its convergence Dr. Noemi Friedman PDE 2 Seite 27

28 Convergence using piecewise higher-order polynomials Theorem: {T h } : non-degenerate family of triangulations of a polygonal domain Ω R 2 u H p+1 u I,p : piecewise d-order approximation There exists a constant C depending on Ω and the value ρ such that [Chapter 5.2] u u I u u I L2 Ch d+1 u Hd+1 H1 Ch d u Hd+1 where: See proof in [3]: Brenner&Scott: The Mathematical Theory of FEM, Chapter 4.4 FEM and its convergence Dr. Noemi Friedman PDE 2 Seite 28

29 Convergence using piecewise higher-order polynomials d = 2 [Chapter 5.2] d = 4 Source: Gockenbach: Understanding and Implementing FEM FEM and its convergence Dr. Noemi Friedman PDE 2 Seite 29

30 Convergence using piecewise higher-order polynomials u u I L 2 Ch d+1 u H d+1 u u I H 1 Ch d u H d+1 u u h E u u I E? convert to homogeneous problem: u u h H 1 M δ u u I H 1 M δ Chd u H d+1 = O(h d ) G:known function,g = g on Γ D u:new function that we look for u = G + u Ω κ(x) u x v x dω = Ω f x v x dω + Γ N hv x dγ Ω κ(x) G x v x dω from natural/neumann BC from essential/dirichlet BC FEM and its convergence Dr. Noemi Friedman PDE 2 Seite 30

31 Convergence using piecewise higher-order polynomials First let s suppose homogenous Dirichlet condition: κ x u x v x dω = Ω Ω f x v x dω + Γ N hv x dγ u u h E u u I E If 0 < k 0 κ k 1 u u I E = a u u I, u u I M u u I H 1 C h d u H d+1 One can also show for inhomogeneous boundary conditions: u u h E 2C h d u H d+1 [Chapter 5.3] FEM and its convergence Dr. Noemi Friedman PDE 2 Seite 31

32 Variational crime: curved boundary [Chapter 5.5] Source: Gockenbach: Understanding and Implementing FEM Variational crimes Céa s lemma and the error estimators may not be valid anymore but: additional errors can be also estimated (Strang) FEM and its convergence Dr. Noemi Friedman PDE 2 Seite 32

33 Variational crime: numerical integration [Chapter 5.5] p(x) p x = ax Strong form: l u 0 = u l = 0 l/5 l/5 l/5 l/5 l/5 Weak form: Discretisation of the weak form: 4 l EA du l dψ dx dx dx = p x ψ x dx 0 0 ψ x u x u i ψ i (x) i=1 K ij f j 4 i=1 u j EA l ψ i (x) x ψ j (x) dx = x l p(x)ψ j (x)dx For more complicated trial functions, p(x) and nonconstant EA(x) difficult to calculate Example: f j = l p x ψ j x dx Numerical integration (example: Gauß-quadrature) n k=1 ω k p x k ψ j x k FEM and its convergence Dr. Noemi Friedman PDE 2 Seite 33

34 Local/ coordinate system, isoparametric mapping 1D x Shape functions: coordinate transformation using the ansatzfunctions Transformation from local to global coordinates: x ξ = x i N 1 ξ + x i+1 N 2 ξ = N 1 ξ Stiffness matrix with isoparametric elements: K 4 e k, l = EA K 4 e k, l = EA ξ = [0,1] N 1 ξ = 1 ξ N 2 ξ = ξ ψ i (x) Ω 4 x 0 1 N k (ξ) ξ local coordinate global coordinate x i x isoparametric mapping 2 ψ j (x) dx = EA x dx dξ 1 Nl (ξ) ξ N 2 ξ ξ = 0 ξ = 1 x i x i+1 N k (ξ) Ω 4 ξ dx dξ 1 dx(ξ) dξ functions of lower order: subparametric functions of higher order: superparametric ξ N l (ξ) ξ x ξ x dx i, j [4,5] 1 k, l [1,2] FEM and its convergence Dr. Noemi Friedman PDE 2 Seite 34 dx dξ 1 dξ dx dξ dx dξ = x dn 1 ξ i dξ dx dξ = dn 1 ξ dξ + x i+1 dn 2 ξ dξ dn 2 ξ dξ x i x i+1

35 Local/ coordinate system, isoparametric mapping 2D triangular elements Basis functions: 1 Transformation from local to global coordinates: x glob ξ, η y glob ξ, η Stiffness matrix: = N 1 ξ, η N 2 ξ, η N 1 ξ, η N 2 ξ, η N 3 ξ, η N 3 ξ, η x 1 y 1 x 2 y 2 x 3 y 3 i, j [1,2,3] FEM and its convergence Dr. Noemi Friedman PDE 2 Seite 35

36 Local/ coordinate system, isoparametric mapping 2D triangular elements Stiffness matrix: i, j [1,2,3] Stiffness matrix with local coordinates: N j N i 1 1 η K ij = J T ξ J N T ξ 0 0 j N i η η where: J dξdη substitution rule determinant should not be negative or zero! i, j [1,2,3] 4 Ni ξ, η 4 Ni ξ, η J = i=1 ξ x i i=1 η x i i=1 4 Ni ξ, η ξ y i i=1 4 Ni ξ, η η y i FEM and its convergence Dr. Noemi Friedman PDE 2 Seite 36

37 Local/ coordinate system, isoparametric mapping 2D triangular elements, example Transformation from local to global coordinates (isoparametric mapping): x glob ξ, η y glob ξ, η x ξ, η y ξ, η = = N 1 ξ, η N 2 ξ, η N 1 ξ, η 1 ξ η ξ 1 ξ η ξ N 2 ξ, η η η N 3 ξ, η N 3 ξ, η FEM and its Convergence Dr. Noemi Friedman PDE 2 Seite 37 x 1 y 1 x 2 y 2 x 3 y 3

38 Local/ coordinate system, isoparametric mapping 2D triangular elements, example Stiffness matrix with local coordinates: K ij = η J T 0 N j ξ N j η J T N i ξ N i η J dξdη i, j [1,2,3] FEM and its convergence Dr. Noemi Friedman PDE 2 Seite 38

39 Condition number of the stiffness matrix Condition number What happens with the roundoff errors in K = LU = K + δk K Ku = f K + δk u = f + δf u u u λ max λ min δf f λ max λ min = κ(k) Condition number of K with nodal bases with 2D triangular mesh: For the Poisson equation N i=1 c i Ψ i (x) Ψ j x dω = Ω f x Ψ j x dω Ω turns il-conditioned for refined mesh!!! K ij FEM and its convergence Dr. Noemi Friedman PDE 2 Seite 39