The Spectral/hp-Finite Element Method for Partial Differential Equations

This set of lecture notes provides an elementary introduction to both the classical Finite Element Method (FEM) and the extended Spectral/$hp$-Finite Element Method for solving Partial Differential Equations (PDEs). Many problems in science and engineering can be formulated mathematically and involves one or more PDEs.

The FEM is nowadays an important numerical discretization technique for approximately solving such mathematical equations on a computer. The set of lecture notes has been written for engineering students for use in the short three-week course {\em 02623 The Finite Element Method for Partial Differential Equations} given at the Technical University of Denmark.

The basic aim of the current lecture notes follows that of the earlier successful lecture notes for the course \cite{BarkerReffstrup1998}, which is to describe the FEM in a way that supports the reader in implementing the method independently. The original set of course notes has been modified and updated and additional chapters describing the high-order extensions to form the Spectral/$hp$-Finite Element Method have been included.

Thus the significant contributions of Chapters 1, 2 and 5 covering the classical Finite Element Method are in large parts due to V. A. Barker and J. Reffstrup. With this set of lecture notes it should be possible for the reader to make a Spectral/$hp$-FEM toolbox in successive steps with the support given in the text.

Emphasis is on the practical details supported with basic and sufficient theory to build the foundation in a three weeks period where the tools are developed and applied immediately. Furthermore, the aim of Spectral/$hp$-FEM toolbox is to provide a simple and generic framework for developing small prototype applications rather than directly approaching large-scale models.

With this in mind, the goal is to let the reader encounter the typical problems involved in the practical implementation of these models and thereby gain a fundamental understanding of the algorithms and their practical implementation. For the practical work, a number of templates described using pseudo programming code, should be understood and converted by the reader to a programming language in a concrete implementation.

A number of exercises is given which in a step-by-step manner guides the reader toward developing the necessary subroutines which can be used to solve typical and fairly general PDEs in one or two spatial dimensions. In the course the chosen programming environment is Matlab, however, this is by no means a necessary requirement.

The mathematical level needed to grasp the details of this set of notes requires an elementary background in mathematical analysis and linear algebra. Each chapter is supplemented with hands-on exercises and the amount of material covered is intended to be balanced in such a way that each subject amounts to approximately one weeks work including producing a small report to document and communicate the work effort.

This set of lecture notes is currently a working draft and may contain some (hopefully) minor errors. Any suggestions for improving the notes or feedback on errors and the content and its structure will be highly appreciated. Please report to the email apek@imm.dtu.dk. \\ \\ \\ Allan P. Engsig-Karup, \today.