Lecture 12 - 12 Galerkin and Ritz Methods for Elliptic PDEs...

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12 Galerkin and Ritz Methods for Elliptic PDEs 12.1 Galerkin Method We begin by introducing a generalization of the collocation method we saw earlier for two-point boundary value problems. Consider the elliptic PDE Lu ( x ) = f ( x ) , (110) where L is a linear elliptic partial differential operator such as the Laplacian L = 2 ∂x 2 + 2 ∂y 2 + 2 ∂z 2 , x = ( x, y, z ) R 3 . At this point we will not worry about the boundary conditions that should be posed with (110). As with the collocation method discussed earlier, we will obtain the approximate solution in the form of a function (instead of as a collection of discrete values). There- fore, we need an approximation space U = span { u 1 , . . . , u n } , so that we are able to represent the approximate solution as u = n j =1 c j u j , u j ∈ U . (111) Using the linearity of L we have Lu = n j =1 c j Lu j . We now need to come up with n (linearly independent) conditions to determine the n unknown coefficients c j in (111). If { Φ 1 , . . . , Φ n } is a linearly independent set of linear functionals, then Φ i n j =1 c j Lu j - f = 0 , i = 1 , . . . , n, (112) is an appropriate set of conditions. In fact, this leads to a system of linear equations Ac = b with matrix A = Φ 1 Lu 1 Φ 1 Lu 2 . . . Φ 1 Lu n Φ 2 Lu 1 Φ 2 Lu 2 . . . Φ 2 Lu n . . . . . . . . . Φ n Lu 1 Φ n Lu 2 . . . Φ n Lu n , coefficient vector c = [ c 1 , . . . , c n ] T , and right-hand side vector b = Φ 1 f Φ 2 f . . . Φ n f . Two popular choices are 128
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1. Point evaluation functionals , i.e., Φ i ( u ) = u ( x i ), where { x 1 , . . . , x n } is a set of points chosen such that the resulting conditions are linearly independent, and u is some function with appropriate smoothness. With this choice (112) becomes n j =1 c j Lu j ( x i ) = f ( x i ) , i = 1 , . . . , n, and we now have an extension of the collocation method discussed in Chapter 9 to elliptic PDEs is the multi-dimensional setting. 2. If we let Φ i ( u ) = u, v i , an inner product of the function u with an appropriate test function v i , then (112) becomes n j =1 c j Lu j , v i = f, v i , i = 1 , . . . , n. If v i ∈ U then this is the classical Galerkin method , otherwise it is known as the Petrov-Galerkin method . 12.2 Ritz-Galerkin Method For the following discussion we pick as a model problem a multi-dimensional Poisson equation with homogeneous boundary conditions, i.e., -∇ 2 u = f in Ω , (113) u = 0 on Ω , with domain Ω R d . This problem describes, e.g., the steady-state solution of a vibrating membrane (in the case d = 2 with shape Ω) fixed at the boundary, and subjected to a vertical force f .
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