IntroductiontoGalerkinMethods - Illinois

Introduction to galerkin MethodsTAM 470 October 19, 20161 IntroductionThese notes provide a brief introduction to galerkin projection methods for numerical solution ofpartial differential equations (PDEs). Included in this class of discretizations are finite elementmethods (FEMs), spectral element methods (SEMs), and spectral methods . A key feature of thesemethods is that they rely on integrals of functions that can readily be evaluated on domains ofessentially arbitrary shape. They thus offer more geometricflexibility than standard finite differenceschemes. It is also easier to develop high-order approximations, where the compact support ofFEM/SEM basis functions avoids the boundary difficulties encountered with the extended stencilsof high-order finite introduce the galerkin method through the classic Poisson problem indspace dimensions, 2 u=fon , u= 0 on .(1)Of particular interest for purposes of introduction will bethe cased= 1, d2 udx2=f, u( 1) = 0.(2)We use uto represent the exact solution to (1) anduto represent our numerical with afinite-dimensionalapproximation spaceXN0and associated set of basis func-tions{ 1, 2.}

Once the requisite properties of the trial/test spaces are identified, the Galerkin scheme is relatively straightforward to derive. One formally generates the system matrix A with right hand side b and then solves for the vector of basis coefficients u. Extensions of the Galerkin method to more complex systems of equations is also straightforward.

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