Aims & Learning Objectives:
Aims: To teach numerical methods for elliptic and parabolic partial differential equations via the finite element method based on variational principles.
Objectives:
At the end of the course students should be able to derive and implement the finite element method for a range of standard elliptic and parabolic partial differential equations in one and several space dimensions. They should also be able to derive and use elementary error estimates for these methods. They should be able to demonstrate an in-depth understanding of the subject.
Content:
Introduction
Variational and weak form of elliptic PDEs. Natural, essential and mixed boundary conditions. Linear and quadratic finite element approximation in one and several space dimensions. An introduction to convergence theory.
System assembly and solution, isoparametric mapping, quadrature, adaptivity.
Applications to PDEs arising in applications.
Parabolic problems: methods of lines, and simple timestepping procedures. Stability and convergence.
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