Analytical and Numerical Methods for Elliptic PDEs
Winterterm 1998/99
The first chapter gives an introduction to elliptic PDEs.
Fundamental properties of harmonic functions are presented
and elementary solution concepts are introduced.
In the first part of the second chapter, the theory of classical solutions is presented
including maximum principles,
uniqueness, continuous dependence on data and existence.
The second part of the second chapter deals with 1D finite difference schemes for
elliptic problems.
The third part continues with 2D finite difference schemes .
The chapter is concluded with some remarks on Green's functions.
In the first part of the third chapter, the
weak formulation of elliptic boundary value problems is introduced
by giving some examples.
After that, the
general Dirichlet problem is investigated and eigenvalue problems
are addressed.
The chapter concludes with a description of the
Finite Element Method .
The first part of the appendix gives a brief review on Numerical Linear Algebra.
The second part introduces basic concepts of generalized functions.
The appendix concludes with a chapter about Sobolev spaces.
The bibliography .
The complete version of the lecture notes.
last changes: 03/04/2000
The Exercises are contained in the lecture notes.
In the book Hilbert Space Methods for
Partial Differential equations
by
Prof. R. E. Showalter
the concept of weak solutions is treated comprehensively.