Advanced Methods of Numerical Analysis 1

March 2008, 4 credits

Introduction

Historical review.
Main approach to the construction of numerical methods for differential equations.
Main sources and parts of the errors contained in approximate solutions.
Challenging problems in modern approximation theory for PDEs.

Mathematical background

Functional spaces, weak derivatives, embedding theorems.
Main classes of boundary-value problems in mahtematical physics.
Existence theory
    Existence theory based on Lax-Milgram lemma.
    Existence theory based on lower semicontinuity of functionals.
General principles of the approximation of functions.
    Projection type error estimates.
    Rate convergence estimates.
    Kolmogorov's n-width.

Approximation methods

Finite difference method.
Variational methods, methods of Ritz, Trefftz, and Kantorovich.
Finite element method.
Integral equations. Iteration methods.
Banach fixed point theorem.
    Applications to linear algebra, ordinary differential equations and integral equations.
Mixed methods I.
    Primal, dual, and mixed formulations for the diffusion problem.
    Existence of solutions, convergence of approximate solutions.
Mixed methods II.
    Dual mixed formulation.
    Inf-sup condition.
    Conforming approximations in the space H(Ω,div), Raviart--Thomas elements.
Mixed methods III.
    Mixed hybrid methods.
    Mixed methods on polygonal cells.
    Convergence of approximations.
    Practical implementation.
Discontinous Galerkin method (DGM)
    DGM for ordinary differential equations.
    DGM for linear elliptic problems.
    DGM for evolutionary problems.
Finite Volume Methods (FVM)

Error control methods

Introduction.
    A review of basic results in approximation theory and a priori asymptotic error estimation..
    Principal features of a posteriori error estimates.
    Classification of a posteriori estimates.
    Literature comments.
A posteriori error estimates based upon fixed point theorems.
    Two-sided error estimates for approximate solutions emerging from the Banach fixed point theorem.
    Applications to linear algebra, ordinary differential equ\ ations and integral equations.
A posteriori error estimates for FEM.
    Estimates based upon estimation of the negative norm of a residual.
    Implicit residual methods.
    Methods based upon gradient averaging (post-processing) of computed solutions.
    A posteriori error estimates in terms of linear functionals.
    Functional a posteriori estimates for linear elliptic problems.