COURSE AIMS AND OBJECTIVES: The aim of the course is to introduce classical and modern methods of solving of linear and nonlinear partial differential equations. Finite element and finite volume methods.
COURSE DESCRIPTION AND SYLLABUS:
1. Finite element method. Method is introduced at the elliptic equation first: variational approximation principle, elemetary proof of the Lax-Milgram, interpolation problem for the Lagrange interpolation. The interpolation estimates are proved in the Sobolev spaces, which are introduced intuitively. Nonconform approximation onlz in the abstract setting of Strang lemmas. Numerical integration, inverse estimates, Aubin-Niche lemma, nonhomogeneous boundary conditions and regular triangulations. Condition number of the approximation problem matrix.
2. Mixed finite element method for difusion. Babuška-Brezzi condition. Raviar-Thomas elements.
3. Parabolic equations. Varijational formulation and apriori estimates explained in a formal procedure. Time discretisation (implicit and explicit) and apriori estimates. Applications: Pironeau's FreeFEM++. Iterative solvers and preconditioning informaly introduced.
4. Finite volume method. Discretization on regular meshes: cell-centred and cell-vertex methods. Applications on eliptic and parabolic equations and the covection-difusion equation. Comparasion with mixed finite element method.
5. During the course at least one nonlinear problem should be solved using the Newton iterations.
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