Review of Matrix Algebra - AIMS Lecture Notes 2006 Peter J...

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Unformatted text preview: AIMS Lecture Notes 2006 Peter J. Olver 14. Finite Elements In this part, we introduce the powerful finite element method for finding numerical approximations to the solutions to boundary value problems involving both ordinary and partial differential equations can be solved by direct integration. The method relies on the characterization of the solution as the minimizer of a suitable quadratic functional. The innovative idea is to restrict the infinite-dimensional minimization principle characterizing the exact solution to a suitably chosen finite-dimensional subspace of the function space. When properly formulated, the solution to the resulting finite-dimensional minimization problem approximates the true minimizer. The finite-dimensional minimizer is found by solving the induced linear algebraic system, using either direct or iterative methods. We begin with one-dimensional boundary value problems involving ordinary differential equa- tions, and, in the final section, show how to adapt the finite element analysis to partial differential equations, specifically the two-dimensional Laplace and Poisson equations. 14.1. Finite Elements for Ordinary Differential Equations. The characterization of the solution to a linear boundary value problem via a quadratic minimization principle inspires a very powerful and widely used numerical solution scheme, known as the finite element method . In this final section, we give a brief introduction to the finite element method in the context of one-dimensional boundary value problems involving ordinary differential equations. The underlying idea is strikingly simple. We are trying to find the solution to a bound- ary value problem by minimizing a quadratic functional P [ u ] on an infinite-dimensional vector space U . The solution u ? ∈ U to this minimization problem is found by solving a differential equation subject to specified boundary conditions. However, minimizing the functional on a finite-dimensional subspace W ⊂ U is a problem in linear algebra, and, moreover, one that we already know how to solve! Of course, restricting the functional P [ u ] to the subspace W will not, barring luck, lead to the exact minimizer. Nevertheless, if we choose W to be a sufficiently “large” subspace, the resulting minimizer w ? ∈ W may very well provide a reasonable approximation to the actual solution u ? ∈ U . A rigor- ous justification of this process, under appropriate hypotheses, requires a full analysis of the finite element method, and we refer the interested reader to [ 45 , 49 ]. Here we shall concentrate on trying to understand how to apply the method in practice. 3/15/06 233 c ° 2006 Peter J. Olver To be a bit more explicit, consider the minimization principle P [ u ] = 1 2 k L [ u ] k 2- h f , u i (14 . 1) for the linear system K [ u ] = f, where K = L * ◦ L, representing our boundary value problem. The norm in (14.1) is typically based on some form of weighted inner product hh v , e v ii on the space of strains...
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Review of Matrix Algebra - AIMS Lecture Notes 2006 Peter J...

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