Numerical methods for PDEs FEM - abstract formulation, the Galerkin method

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1 Patzhater für Bid, Bid auf Titefoie hinter das Logo einsetzen Numerica methods for PDEs FEM - abstract formuation, the Gaerkin method Dr. Noemi Friedman

2 Contents of the course Fundamentas of functiona anaysis Abstract formuation FEM Appication to conrete formuations Convergence, reguarity Variationa crimes Impementation Mixed formuations (e.g. Stokes) Error indicators/estimation Adaptivity Gaerkin method Dr. Noemi Friedman PDE2 tutoria Seite 2

3 Abstract formuation, exampes Introduction From strong form to weak form Weak form of BVPs with inhomogenous BCs Best approximation by orthogona projection Orthogona projection Gaerkin method, minimized error for symmetric BVPs Bound of error of Gaerkin method (Céa s theorem) Gaerkin method Dr. Noemi Friedman PDE2 tutoria Seite 3

4 Introduction to Gaerkin method Matab code from Gockenbach book: Equations foowed throughout the semester [Chapter1]: Lapace/Poisson equation: bar with uniaxia oad steady state heat fow sma vertica defections of a membrane stationary heat equation Other eptic BVPs: isotropic easticity Gaerkin method Dr. Noemi Friedman PDE 2 Seite 4

5 FROM STRONG FORM TO WEAK FORM 1D: Steps of formuating the weak form (recipe) [Chapter 2] Lu(x) = f(x) 1.) Mutipy by test/weight function v(x) and integrate Lu, v p, v = v Lu x v x dx f x v x dx = v Check wether the PDE hods in the v weighted avarage sense over 2.) Reduce order of Lu, v by using the integration by parts 3.) Appy boundary conditions if it hods for a test functions from a sufficienty arge set, then the PDE must hod [Gockenbach] Dr. Noemi Friedman PDE 2 Seite 5

6 1D Exampe: bar under axia oads with inhomogenious Neumann BC. [On inhomogenous BCs see Chapter 2.2] I. Formuate the strong equation 1.) kinematica equation: (strain dispacement) p(x) t 2.) equiibrium equation: (forces in equiibrium) 3.) constitutive equation: (stress strain) 4.) Boundary conditions: u = Dirichet boundary u = Aσ = t Neumann boundary EA du /dx = t x = [, ] Dr. Noemi Friedman PDE 2 Seite 6

7 1D Exampe: bar under axia oads with inhomogenious Neumann BC. inear term f(v) : p, v = Lu, v = biinear term a(u, v): p x v x dx Lu, v p, v = v H 1 Integration by parts: EA d2 u(x) dx 2 v(x)dx a a = EA du b = ab ab b = dv dx dx Lu, v = EA du dx v Lu, v p, v = EA du dv dx dx dx = tv() + EA du dv dx dx dx v = EA du dv dx dx dx tv p x v x dx = EA du dx x= = t EA du dv dx dx dx = p x v x dx + tv() (principe of virtua work) Dr. Noemi Friedman PDE 2 Seite 7

8 Mutidimensiona stationary heat equation with inhomogeneous Neumann BC. Strong form: Exampe: Lu(x) = f(x) Δu x = f u = u n = h 1.) Mutipy by test function v and integrate Δu x v x d f x v x d = 2.) Reduce biinear term s order by using Green s identity a u, v = Lu, v : Δu x v x d = u x v x d u v x dγ n. Dr. Noemi Friedman PDE 2 Seite 8

9 Mutidimensiona stationary heat equation with inhomogeneous Neumann BC. Δu x v x d = u x v x d u v x dγ n 3.) Appy boundary conditions u n v x dγ = u Γ N n v x dγ + u Γ D n v x dγ = hv x dγ Γ N h Δu x v x d = u x v x d hv x dγ Γ N u x v x d = f x v x d + Γ N hv x dγ. Dr. Noemi Friedman PDE 2 Seite 9

10 Mutidimensiona stationary heat equation with inhomogeneous Dirichet and Neumann BC. Strong form: Exampe: Lu(x) = f(x) Δu x = f u = g u n = h 1.) Mutipy by test function v and integrate convert to homogeneous probem: u = ω + u ω:known function,ω = g on Γ D u:new function that we ook for Δu x v x d f x v x d = 2.) Reduce order of Lu, v by using divergence theoreem Δu x v x d = u x v x d u v x dγ n. Dr. Noemi Friedman PDE 2 Seite 1

11 Mutidimensiona stationary heat equation with inhomogeneous Dirichet and Neumann BC. Δu x v x d = u x v x d u v x dγ n 3.) Appy boundary conditions u n v x dγ = u Γ N n v x dγ + u Γ D n v x dγ = hv x dγ Γ N h u x v x d = f x v x d + Γ N hv x dγ ω x + u x v x d = f x v x d + Γ N hv x dγ u x v x d = f x v x d + hv x dγ Γ N ω x v x d from natura/neumann BC from essentia/dirichet BC Gaerkin method Dr. Noemi Friedman PDE 2 Seite 11

12 Existence and uniqueness of the soution of BVPs [Chapter 2.4] Strong form: Weak form: Lu(x) = f(x) Lu, v = f, v v? u? a(u, v) F(v) biinear term inear term Does the soution exists? Does it have a unique soution? In accordance to the Lax-Migram Lemma if: F( ) bounded, inear functiona a, bounded, V-eiptic biinear functiona and V a Hibert space For a specific BVP one has to find the right Hibert space (Soboev space or L2 space) where the conditions for a, and F( ) are satisfied, and then we know, that in that space we have a unique soution! Gaerkin method Dr. Noemi Friedman PDE 2 Seite 12 soution u exists unique soution u V +soution u depends continiousy on f

13 Existence and uniqueness of the soution of BVPs exampes 1. Poisson equation: Δu x = f x F( ) inear functiona, in H 1 : bounded a, biinear functiona, in H : bounded, V-eiptic 1 a, biinear functiona, in L 2 : not bounded a, biinear functiona, in H 1 : not V-eiptic 2. Pate equation F( ) inear functiona, in H E 2 : bounded a, ΔΔu x = f x biinear functiona, in H E : bounded, V-eiptic 2 F(v) = f x v x dx a, = u x v x dx F(v) = f x v x dx Gaerkin method Dr. Noemi Friedman PDE 2 Seite 13 unique soution u H 1 v H 1 a, = Δu x Δv x dx we have to narrow down the space in which we ook for the soution, because we can not prove that there is a unique soution in L2 or in H1 unique soution u H E 2 v H E 2

14 Proxi mode projection of the soution [Chapter 3] Let s say we know that there is a unique soution in V = H 1. But V is an infinite dimensiona space Let s narrow down the space, where we are trying to find the soution, to a finite dimensiona space exampe: instead of finding u(x) H 1 we try to find the coefficients α j of a proxi mode (ansatz function): u h x = j=1 Gaerkin method Dr. Noemi Friedman PDE 2 Seite 14 n α j ω j (x) where ω j x : known (ineary independent) basis or ansatz functions u h x : the approximation of the soution u(x), which is in an n-dimensiona space: u h V h = span{ω 1, ω 2, ω n }

15 Proxi mode projection of the soution Let s fix this subspace to a specific V h (that is, we fix the ansatz functions in our exampes) How do we get the best approximation of the soution in this space from the equation: Our goa is to minimize the difference in between the soution and the approximation: error = u x u h x < u x z x z x V h The best approximation u h to u from V h is the one where the error is orthogona to the space of V h, that is to a possibe z x V h. Instead of writing it for a z (as z is an n-dimensiona space) we can write ω i i = 1.. n u x u h x, ω i x = i = 1.. n Lu(x) = f(x) Best approximation to u x from V h Gaerkin method Dr. Noemi Friedman PDE 2 Seite 15

16 Proxi mode projection of the soution Pugging in the proxi mode to the orthogonaity condition we have: u n j=1 α j ω j, ω i = i = 1.. n Rearranging the equation we get: u, ω i n α j ω j, ω i = i = 1.. n j=1 n j=1 α j ω j, ω i = u, ω i i = 1.. n G ij b j n j=1 α j G ij = b j Gα = b Gaerkin method Dr. Noemi Friedman PDE 2 Seite 16

17 Proxi mode projection of the soution Gα = b where G ij = ω j, ω i : can be cacuated from the basis functions and the inner product of the given space (if the basis is orthonorma the matrix is the identity matrix b j = u, ω i :? α j : The coefficients that we are ooking for But how do we get b j = u, ω i? We know Lu, v L2 = f, v L2 If the eft hand side of the weak equation is a V-eiptic, bounded biinear functiona, that is aso symmetric, then it can be written as: Lu, v L2 = a(u, v) = u, v E Gaerkin method Dr. Noemi Friedman PDE 2 Seite 17

18 Proxi mode projection of the soution That means that with Gaerkin method we orthogonaize the projection, that is, we minimize the error in the energy space. And the matrix equation, that we can cacuate the coefficients from, wi have the form: Gα = b with G ij = ω j, ω i E = a(ω j, ω i ) b j = u, ω i E = f, ω i L2 Exampe: x =,, u =, u = Lu = EA 2 u(x) 2 x G ij = ω j (x), ω i (x) E = f = p EA dω j(x) dx dω i (x) dx dx b j = f, ω i L2 = p x ω i x dx Gaerkin method Dr. Noemi Friedman PDE 2 Seite 18

19 Gaerkin method is an orthogona projection The soution of the BVP satisfies the initia weak formuation: a u, v = F v v V and as V h V, it aso satisfies a u, v = F v v V h Simiary, the approximation of the soution u h satisfies: a u h, v = F v v V h Subtracting the ast two equations : a u, v a u h, v = v V h a u u h, v = v V h Gaerkin method gives the best approximation of the true soution in the given subspace V h in the energy norm In case a(u, u) is a symmetric biinear term Gaerkin method Dr. Noemi Friedman PDE 2 Seite 19

20 Céa s theorem Concusion from before: a, : symmetric Gaerkin method gives the best approximation in the energy norm But what about the error in other norms? According to Céa s theorem (see prove at the ecture), even without a, being symmetric, the error of the approximation of Gaerkin wi be aways bounded: u u h M δ u v v V h Where M and δ are constants from the conditions of boundedness and V-eipticity of the biinear term a, : a u, v M u v a u, u δ u 2 and u v is the norm of the difference in between the true soution and any v V h This term depends on the n-dimensiona space V h chosen, and the space where the true soution ies in. Gaerkin method Dr. Noemi Friedman PDE 2 Seite 2

Introduction to PDEs and Numerical Methods Tutorial 10. Finite Element Analysis

Patzhater für Bid, Bid auf Titefoie hinter das Logo einsetzen Introduction to PDEs and Numerica Methods Tutoria. Finite Eement Anaysis Dr. Noemi Friedman, 3..2 FROM STRONG FORM TO WEAK FORM inhomogeneous