This course unit covers the theory of approximation and applications to the numerical solution of linear elliptic partial differential equations (PDEs) using finite element approximation methods. Such methods are universally used to solve practical problems associated with physical phenomena in complex geometries. The emphasis is on assessing the accuracy of the approximation using a priori and a posteriori error estimation techniques. Practical issues will be illustrated with MATLAB using the IFISS software toolbox.
1.Basics. Review of basic functional analysis concepts: norms, inner-products. Sobolev spaces. Weak derivatives. Lax-Milgram lemma. 
2.Linear approximation. Best approximation in Lp norms. Existence and uniqueness. Choice of norm in practical curve fitting. Least squares approximation and normal equations. Orthogonal basis functions. Overview of the cases p=1 and p=. Choice of linear approximating functions. Polynomials, orthogonal polynomials, Chebyshev polynomials, Spline functions. Surface fitting by polynomials and splines, including the thin plate spline and radial basis functions.
3.Finite element methods for the diffusion equation. Affine mappings. Linear, bilinear, quadratic and biquadratic approximation. Finite element assembly process. Properties of the discrete equation system. A priori error bounds: best approximation in energy, H1 error bounds. H2 regularity and singular problems. A posteriori error bounds. Local error estimators. Self adaptive refinement strategies. 
4.Finite element methods for the convection-diffusion equation. Well-posedness. Weak formulation. Galerkin approximation. The streamline-diffusion method. A priori and a posteriori error bounds. Self-adaptive refinement strategies for resolving layers.