TIES595 Numerical Analysis of PDEs (JSS28) (5 cr)

Content

The lecture course is intended to give an overview on mathematical models and methods based on partial differential equations. It consists of the following parts. 1. Introduction. Historical and literature overview. Main problems in qualitative and quantitative analysis of models based on differential equations: existence, stability, convergence of approximations, a priori and a posteriori estimates. 2. Linear elliptic problems. Correctness and approximation. Energy methods. Classical finite element and finite difference methods. Mixed and dual mixed methods. Finite volume method. Discontinuous Galerkin method. Convergence of approximations and rate convergence estimates. Adaptive numerical methods and error indicators. Applications to models of diffusion, elasticity and linear viscous fluids. 3. Nonlinear variational problems in mechanics and physics. Existence of solutions. Variational inequalities and free boundary problems. Numerical analysis of nonlinear problems: regularization and saddle point algorithms. Applications to problems with obstacles, nonlinear viscous fluids, plasticity. 4. Reliable numerical methods and a posteriori error estimates. Main classes of a posteriori error estimators: residual, hierarchical, post processing, and goal-oriented. Functional methods of a posteriori error control. Practical implementation of different methods to finite element approximations.