We begin with an introduction to the concept of finite elements and the Ritz-Galerkin method for the solution of elliptic PDEs typically arising from problems in mechanics. We will derive the basic error estimates using Cea's lemma and treat nonsymmetric/indefinite problems using the approach of Schatz. We will discuss nonconforming methods and mixed methods, and their analysis based on the Berger-Scott-Strang lemma and the L-B-B inf-sup condition. We will then survey discontinuous Galerkin methods for elliptic problems and present a new analysis of discontinuous finite element methods that combines techniques from a priori analysis and a posteriori analysis. After this we will turn to efficient algorithms for solving finite element equations, such as multigrid methods and domain decomposition methods. If time allows, we will also discuss finite element methods for the time-harmonic Maxwell's equations.

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Solutions (by Richard Norton)