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 Application to conrete formulations Convergence, regularity Variational crimes Implementation Mixed formulations (e.g. Stokes) Stabilisation for flow problems Error indicators/estimation Adaptivity Convergence Dr. Noemi Friedman PDE2 tutorial Seite 2

3 Abstract formulation, examples Non-degenerate triangulation, refinement of triangulation Convergence using piecewise linear functions Convergence using piecewise higher order elements Variational crime: numerical integration Variational crime: curved boundaries FEM and piecewise polynomials Implementation, sparsity of the stiffnes matrix Convergence Dr. Noemi Friedman PDE2 tutorial 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 Convergence Dr. Noemi Friedman PDE tutorial Seite 4

5 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. Convergence Dr. Noemi Friedman PDE tutorial Seite 5

6 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: n u I 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. Convergence Dr. Noemi Friedman PDE tutorial Seite 6

7 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 LL Ch 2 u HH u u I HH Ch u HH where: (seminorm) Convergence Dr. Noemi Friedman PDE tutorial Seite 7

8 Convergence using piecewise linear functions [Chapter 5.1] Source: Gockenbach: Understanding and Implementing FEM Convergence Dr. Noemi Friedman PDE tutorial Seite 8

9 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 LL Ch d+1 u HH+1 u u I HH Ch d u Hd+1 where: Convergence Dr. Noemi Friedman PDE tutorial Seite 9

10 Convergence using piecewise higher-order polynomials d = 2 [Chapter 5.2] d = 4 Source: Gockenbach: Understanding and Implementing FEM Convergence Dr. Noemi Friedman PDE tutorial Seite 10

11 Variational crime: numerical integration [Chapter 5.5] p(x) p x = aa Strong form: l u 0 = u l = 0 l/5 l/5 l/5 l/5 l/5 Discretisation of the weak form: 4 u x u i ψ i (x) i=1 Weak form: l EE dd dd 0 dd dd dd l = p x ψ x dd 0 φ x K ij f j 4 u j EE ψ i(x) ψ j (x) dx i=1 l = p(x)ψ j (x)dx l For more complicated trial functions, p(x) and nonconstant EA(x) difficult to calculate Example: f j = p x ψ j x dd l Numerical integration (example: Gauß-quadrature) n ω k p x k ψ j x k k=1 Convergence Dr. Noemi Friedman PDE tutorial Seite 11

12 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) Convergence Dr. Noemi Friedman PDE tutorial Seite 12

13 Piecewise polynomials and the FEM [Chapter 4] N c i i=1 Ψ i (x) Ψ j x dω Ω = f x Ψ j x dω Ω KK = 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 Convergence Dr. Noemi Friedman PDE tutorial Seite 13

14 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 = u i ψ i x continious piecewise linear functions for Poisson equation we have to show that P h 1 H 1 Convergence Dr. Noemi Friedman PDE tutorial Seite 14 4 i=1 u h (x) P h 1

15 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 Convergence Dr. Noemi Friedman PDE tutorial Seite 15

16 Sparsity of the stiffness matrix [Chapter 4.1] K ii = 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 Convergence Dr. Noemi Friedman PDE tutorial Seite 16

17 Quadratic piecewise polynomials Ψ i x, y = Ψ 5 Ψ 18 [Chapter 4.2] 6 nodes needed Convergence Dr. Noemi Friedman PDE tutorial Seite 17

18 Quadratic piecewise polynomials [Chapter 4.2] nn = = 16% nn = = 5% Convergence Dr. Noemi Friedman PDE tutorial Seite 18

19 Cubic piecewise polynomials [Chapter 4.3] Convergence Dr. Noemi Friedman PDE tutorial Seite 19

20 Higher order piecewise polynomials [Chapter 4.4] 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 d = 4 d = 6 d = 5 Convergence Dr. Noemi Friedman PDE tutorial Seite 20