Example: bachelor of science

IntroductiontoGalerkinMethods - University of 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.

general Neumann or Robin boundary conditions, which is not generally the case for finite difference methods. 3. Deriving a System of Equations We develop (6) into a discrete system appropriate for computation by inserting the expansions v = P i vi ...

Tags:

  Conditions, Robin, Boundary, Robin boundary conditions

Information

Domain:

Source:

Link to this page:

Please notify us if you found a problem with this document:

Other abuse

Transcription of IntroductiontoGalerkinMethods - University of Illinois ...

1 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.

2 (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, .. , n} XN0satisfying the homogeneous boundary condition i= 0 on , thestandard approach to deriving a Galerkin scheme is to multiply both sides of (1) by a test functionv XN0, integrate over the domain, and seek a solutionu(x) :=Puj j(x) satisfying Z v 2u dV=Z vf dV v XN0.(3)The Galerkin scheme is essentially a method of undeterminedcoefficients. One hasnunknownbasis coefficients,uj,j= 1, .. , nand generatesnequations by successively choosing test functionsvthat spanXN0( ,v= i,i= 1.)

3 , n). Equating both sides for every basis function inXN0ensures that the residual,r(x;u) := 2u fis orthogonal toXN0, which is why these methodsare also referred to weighted residual theL2inner product (f, g) :=R f g dV, (3) is equivalent to findingu XN0for which(v, r) :=Z v r(x, u)dV= 0, v XN0.(4)That is,r(u) is orthogonal tovor, in this case, the entire space:r XN0. Convergence,u u,is achieved by increasingn, the dimension of the approximation space. As the space is completed,the only function that can be orthogonal to all other functions is the zero function, such thatu is important to manipulate the integrand on the left of (3)to equilibrate the continuityrequirements onuandv. Integrating by parts, one has Z v 2u dV=Z v u dV Z v u ndA(5)The boundary integral vanishes becausev= 0 on (v XN0) and the Galerkin formulation reads:Findu XN0such thatZ v u dV=Z vf dV v XN0.

4 (6)Note that the integration by parts serves to reduce the continuity requirements onu. We need onlyfind authat is once differentiable. That is,uwill be continuous, but uneed not be. Of course, if uis continuous, then uwill converge to ufor a properly formulated and implemented (6) is the point of departure for most finite element, spectral element, and spectralformulations of this problem. To leading order, these formulations differ primarily in the choice ofthe is and the approach to computing the integrals, both of which influence the computational ef-ficiency for a given problem. The Galerkin formulation of thePoisson problem has many interestingproperties. We note a few of these here: We ve not yet specified requisite properties ofXN0, which is typically the starting point (andoften the endpoint) for mathematical analysis of the finite element method.

5 Two functionspaces are of relevance to us:L2, which is the space of functionsvsatisfyingR v2dV < ,andH10, which is the space of functionsv L2satisfyingR v v dV < andv= 0 on . OverL2, we have the inner product (v, w) :=R vw dVand norm||v||:=p(v, v). For anyv, w H10, we also define the energy inner producta(v, w) :=R v w dVand associatedenergy norm,||w||a:=pa(w, w). For the (continuous) Galerkin method introduced here, wetakeXN0 H10. A key result of the Galerkin formulation is that, over all functions inXN0,uisthebest fitapproximation to uin theenergy norm. That is:||u u||a ||w u||a w XN0. The best fit property is readily demonstrated. For anyv, w H10, one can show that(v, r(w)) =a(v, w u).

6 Defining the errore:=u uand using the orthogonality of theresidualr(u), one has0 = (v, r(u)) =a(v, e) v XN0 That is, the error is orthogonal to the approximation space in theainner product,e depicted in Fig. 1,uis the closest element ofXN0to a XN0eu uFigure 1:a-orthogonal projection of uontoXN0. Foru XN0, andv YN0we refer toXN0as thetrial spaceandYN0as the test that the cardinalities ofXN0andYN0are equal, the spaces need not be the particular, if one choosesYN0=span{ (x x1), (x x2), .. , (x xn)}one recovers thestrong formin which (1) is satisfied pointwise. The Galerkin statement (6) is often referredto as theweak form, the variational form, or the weighted residual form.

7 The variational form (6) leads to symmetric positive definite system matrices, even for moregeneral Neumann or robin boundary conditions , which is not generally the case for finitedifference a System of EquationsWe develop (6) into a discrete system appropriate for computation by inserting the expansionsv=Pivi iandu=Pjuj jinto the integrand on the left of (6) to yieldZ nXi=1vi i(x)! nXj=1uj j(x) dV=nXi=1nXj=1vi Z i(x) j(x)dV uj(7)Equating (46) to the right side of (6) and definingAij:=Z i(x) j(x)dV(8)bi:=Z i(x)f dV(9)v:= (v1, v2, .. , vn)T(10)u:= (u1, u2, .. , un)T(11)the discrete Galkerin formulation becomes,Findu lRnsuch thatnXi=1nXj=1viAijuj=:vTAu=vTb v lRn,(12)or, equivalently:Au=b.

8 (13)This is the system to be solved for the basis coefficientsui. It is easy to show thatAis symmetricpositive definite (xTAx>0 x6= 0) and therefore invertible. TheconditioningofA, however,is not guaranteed to be amenable to finite-precision computation unless some care is exercised inchoosing the basis for approximation. Normally, this amounts merely to finding is that are nearlyorthogonal or, more to the point, far from being linearly dependent. We discuss good and bad basischoices shortly. (Generally speaking, one can expect to lose log10 (A) digits due to round-offeffects when solving (13).)ExtensionsOnce the requisite properties of the trial/test spaces are identified, the Galerkin scheme isrelatively straightforward to derive.

9 One formally generates the system matrixAwith right handsideband then solves for the vector of basis coefficientsu. Extensions of the Galerkin methodto more complex systems of equations is also straightforward. For the example of the reaction-convection-diffusion equation, 2u+c u+ 2u=f, the procedure outlined above leadsto Au+Cu+ 2Bu=b,(14)withCij:=R ic jdVandBij:=R i jdV. We refer toAas the stiffness matrix,Bthemass matrix, andCthe convection operator. For the convection dominated case( , small, = 0), one must be judicious in the choice of trial and test spaces. Another important extensionis the treatment of boundary conditions other than the homogeneous Dirichlet conditions (u= 0on ) considered so far.

10 In addition, development of anefficientprocedure requires attention tothe details of the implementation. Before considering these extensions and details, we introducesome typical examples of bases ExamplesWe consider a few examples of 1D basis functions. In additionto the trial/test spaces and associatedbases, we will introduce a means to compute the integrals associated with the systems matrices,A,B, andC. We have the option of using exact integration or inexact quadrature. In the lattercase, we are effectively introducing an additional discretization error and must be mindful of thepotential functions (in any space dimensiond) come in essentially two forms, basis functions are known as Lagrangian interpolantsand have the property that the basiscoefficientsuiare alsofunction valuesat distinct pointsxi.


Related search queries