Generalized Finite Element Methods Other approximation methods Suvranu De

Approximation properties of the Galerkin method Last class Approximation properties of the Galerkin method

Last class Increased refinement Galerkin method Properties Property 1: The discretization error is orthogonal to the approximation space Xh in the energy norm Property 2: Strain energy of mathematical model Strain energy of discretized model Increased refinement

Last class Galerkin method Properties Property 3: Best approximation property

Other approximation techniques Point collocation method This class Other approximation techniques Point collocation method Subdomain integral/average error method Weighted residual method (GUSWF I) Petrov-Galerkin method (GSWF) H-1 method (GUSWF II)

Convection-Diffusion Motivation given functions of (x,y) Scalar, Linear, Parabolic equation

Convection-Diffusion Motivation Applications If u(x,y) is.... Temperature  Heat Transfer Pollutant concentration  Coastal Engineering Probability distribution  Statistical Mechanics Price of an option  Financial Engineering ....

Convection-Diffusion Motivation 1D example... 1D steady state fluid flow in a channel with heat transfer x=0 x=L v qL qR x Notice Elliptic Very close to the Poisson problem except for an additional derivative of odd order.

Convection-Diffusion Motivation 1D example... The BVP

Convection-Diffusion Motivation 1D example... The VBVP

Convection-Diffusion Motivation 1D example... Solution using Galerkin method shows spurious oscillations

Model problem BVP The strong formulation (BVP) Domain boundary Find u  X such that subject to boundary conditions u may be a scalar or a vector (e.g., elasticity/fluid flow in 2/3D) The operators A and B can be scalar/vector. The problem may be posed in 1/2/3 D

Approximation XhX Residuals Define a finite dimensional subspace Xh of X spanned by linearly independent functions “trial/basis functions” Assume Domain residual Boundary residual

Point collocation method Scheme Point collocation method Make the domain and boundary residuals vanish at N points Domain residual Boundary residual

Point collocation method Example Point collocation method

Point collocation method Example Point collocation method Analytical solution

Point collocation method Example Point collocation method Numerical solution Check Satisfies the essential BC automatically (no need to consider boundary residual) Domain residual

Point collocation method Example Point collocation method Numerical solution In the point collocation method we pick any point x=x1 in the domain and set the domain residual to zero at that point Domain residual

Point collocation method Example Point collocation method Numerical solution For x1<0.5 Soln symmetric about x=0.5 The max overshoots As x1  0, (uh)max 

Point collocation method Example Point collocation method Numerical solution For x1>0.5, f(x1)=0, hence uh(x)=0 !!! Solution very sensitive to the choice of collocation point

Point collocation method Example Point collocation method Analytical solution Satisfies the essential BC automatically (no need to consider boundary residual) What happens if x1 and x2 are both greater than 0.5? x1  x2 ?

Point collocation method Example Point collocation method Numerical solution For x1=0.2, x2=0.8 Right bias The max overshoots

Point collocation method Comments Point collocation method Numerical solution Solution sensitive to “proper” choice of collocation points. should not be too close should be enough points on the boundary The coefficient matrix is nonsymmetric Simple and rapid Domain residual may not be small in between nodal points A smooth trial function space is required (in this example, need C1 otherwise the second derivatives would not exist) The bandwidth of the matrix depends on the support of the approximation functions

Subdomain collocation method Scheme Subdomain collocation method assume that the satisfy all the boundary conditions Domain residual Subdivide the domain into nonoverlapping subdomains and set the integral of the domain residual to zero over each subdomain. W Wi

Subdomain collocation method Example Subdomain collocation method

Subdomain collocation method Example Subdomain collocation method Numerical solution Satisfies the essential BC automatically (no need to consider boundary residual) Domain residual Integral of domain residual

Subdomain collocation method Example Subdomain collocation method Numerical solution Soln symmetric about x=0.5 The max overshoots (not as much as point collocation).

Subdomain collocation method Example Subdomain collocation method Analytical solution Satisfies the essential BC automatically (no need to consider boundary residual) Subdivide domain into 2 equal intervals Domain residual Set integral of domain residual to zero on each subdomain

Subdomain collocation method Example Subdomain collocation method Numerical solution

Subdomain collocation method Comments Subdomain collocation method Numerical solution Behavior of numerical solution much better than point collocation The coefficient matrix is nonsymmetric More complex than point collocation since integrals have to be evaluated A smooth trial function space is still required (in this example, need C1) It is better to set the integral of the residual to zero than just the residual to zero at a few collocation points. The bandwidth of the matrix depends on the size of the min(size of subdomain, support of approximation function)

Subdomain collocation method Comments Reducing the constraint Subdomain collocation method Less smooth trial functions Numerical solution 1 h (i-1)h ih Wi where 1. N equations in N unknowns 2. Requires the trial functions to be in C1

Subdomain collocation method Comments Reducing the constraint Subdomain collocation method Less smooth trial functions Numerical solution 1 h (i-1)h ih Wi Integrate by parts Same answer as before Requires the trial functions to be in C0 ! Starting point for “finite volume” methods (FVM)

Reducing the constraint Comments Subdomain collocation method 2/3D Numerical solution Wi W Gu Gq n Gi Poisson’s equation Subdomain integral over Wi Use Green’s theorem

Reducing the constraint Comments Subdomain collocation method 2/3D Numerical solution Needs the trial functions to be C0

Least squares method Scheme assume that the satisfy all the boundary conditions Domain residual Minimize the L2 norm of the residual on the domain

Example Least squares method

Least squares method Example Numerical solution Domain residual L2 norm of residual Obtain ith equation by setting

Example Least squares method Numerical solution Symmetric matrix

Least squares method Example Numerical solution Satisfies the essential BC automatically (no need to consider boundary residual)

Least squares method Example Numerical solution Soln symmetric about x=0.5 The max undershoots (unlike point/subdomain collocation).

Least squares method Example Analytical solution Satisfies the essential BC automatically (no need to consider boundary residual)

Example Least squares method Numerical solution Right bias

Least squares method Comments Numerical solution Behavior of numerical solution much better than point / subdomain collocation The coefficient matrix is symmetric and positive definite A smooth trial function space is still required (in this example, need C1) The bandwidth of the matrix depends on the support of the approximation functions

Weighted residual method Scheme Comments Weighted residual method GUSWF I Numerical solution Domain residual assume that the satisfy all the boundary conditions Set the weighted integral of the residual to zero over the entire domain. Global unsymmetric weak form I (GUSWF I)

Weighted residual method Scheme Comments Weighted residual method GUSWF I Numerical solution Choose Yh: Test function space yj(x) : Test function Xh: Trial function space jj(x) : Trial function N conditions to determine the N unknown uhj s

Weighted residual method Scheme Comments Weighted residual method GUSWF I Numerical solution Note yi(x)= d(x-xi): POINT COLLOCATION : SUBDOMAIN COLLOCATION

Petrov-Galerkin method Scheme Comments Petrov-Galerkin method GUSWF I Numerical solution Start from GUSWF I with distinct trial and test function spaces e.g., Wi W Gu Gq n Gi Poisson’s equation GUSWF I : Find uh  Xh such that

Petrov-Galerkin method Comments Scheme Petrov-Galerkin method GSWF Numerical solution GUSWF I requires the trial functions to be at least C1 but the test functions can be C0 (unsymmetry!) To reduce the smoothness requirement on the trial functions by using Green’s theorem Global symmetric weak form (GSWF) Find uh  Xh such that

Petrov-Galerkin method Comments Scheme Petrov-Galerkin method GSWF Numerical solution Choose trial functions jj (and therefore uh) to satisfy the prescribed Dirichlet BC Choose the test functions to vanish on the Dirichlet boundary to get rid of the boundary integral on Gu Find uh  Xh such that

Petrov-Galerkin method Scheme Comments Petrov-Galerkin method GSWF Numerical solution

Petrov-Galerkin method Scheme Comments Petrov-Galerkin method GSWF Numerical solution Since Rewrite GSWF Find uh  Xh such that where

Petrov-Galerkin method Comments Scheme Petrov-Galerkin method GSWF Numerical solution Find uh  Xh such that Note: The Neumann BC is incorporated in the GSWF The Dirichlet BC is satisfied by the trial functions jj The test functions yj vanish on the Dirichlet boundary If Xh=Yh then we revert back to the Galerkin formulation

Example Petrov Galerkin

Petrov-Galerkin method Comments Scheme Petrov-Galerkin method Example Numerical solution GUSWF I GSWF

Petrov-Galerkin method Scheme Comments Petrov-Galerkin method Example Numerical solution

Petrov-Galerkin method Example Petrov-Galerkin method Numerical solution Satisfies the essential BC automatically Choose “Galerkin” choice

Petrov-Galerkin method Example Petrov-Galerkin method Numerical solution Soln symmetric about x=0.5 The max undershoots (unlike point/subdomain collocation).

Petrov-Galerkin method Example Petrov-Galerkin method Analytical solution Satisfies the essential BC automatically Choose Symmetric matrix

Petrov-Galerkin method Example Petrov-Galerkin method Numerical solution

Comparison Example Numerical solution With two functions Point collocation Subdomain collocation Galerkin

H-1 Method Scheme Comments GUSWF II Numerical solution GUSWF I : the trial functions at least C1 , the test functions can be C0 (unsymmetric!) GSWF : the trial functions and test functions at least C0 (symmetric!) GUSWF II : the trial functions at least C0 and test functions at least C1 (unsymmetric!) How? use Green’s theorem one more time

Example H-1 Method

H-1 Method Comments Scheme Example Numerical solution GUSWF I GSWF GUSWF II

Scheme Comments H-1 Method Example Numerical solution

H-1 Method Example Numerical solution Satisfies the essential BC automatically Choose

H-1 Method Example Numerical solution Soln symmetric about x=0.5 The max undershoots (unlike point/subdomain collocation).

H-1 Method Scheme Comments GUSWF II Numerical solution GSWF Find uh  Xh such that GUSWF II Find uh  Xh such that

Summary GUSWF II : the trial functions at least C0 and test functions at least C1 (unsymmetric!) Incorporates both natural and essential BCs Starting point for the Boundary Element Method (BEM)

Point Collocation Special cases Summary GUSWF I : the trial functions at least C1 , the test functions can be C0 Point Collocation Subdomain collocation GSWF (Petrov-Galerkin) : the trial functions and test functions at least C0 Special case: Galerkin method: The trial and test functions are identical GUSWF II : the trial functions at least C0 and test functions at least C1 Special cases

Multiresolution analysis: Wavelet Galerkin Orthogonal Discrete Wavelet Transform or DWT Orthogonal IDWT

In operator form Define Multiresolution analysis: Wavelet Galerkin Orthogonal Discrete Wavelet Transform or DWT Define Orthogonal IDWT

In operator form Multiresolution analysis: Wavelet Galerkin Orthogonal Discrete Wavelet Transform or DWT Orthogonal IDWT

Single scale wavelet Galerkin Multiresolution analysis: Wavelet Galerkin Single scale wavelet Galerkin Find

Two-scale wavelet Galerkin Multiresolution analysis: Wavelet Galerkin Two-scale wavelet Galerkin Premultiply by W

Hence Multiresolution analysis: Wavelet Galerkin Split the single scale equation into 2-scale equation

Multiresolution analysis: Wavelet Galerkin Single scale VBVP Find such that Multiple scale VBVP Find such that