5 Overview

In brief, Femlisp has the following features:

• Unstructured grids in arbitrary space dimensions n \geq 0. Cells can be arbitrary products of simplices.
• Isoparametric and non-parametric element mappings
• Local grid refinement
• Lagrange finite elements of arbitrary order
• Several types of PDEs are already implemented:
  • Convection-diffusion-reaction equations
  • Elasticity
  • Navier-Stokes
• Time-dependent problems using Rothe's method
• Geometric and algebraic multigrid methods
• Graphics (using Data Explorer, Gnuplot)
• Interactive environment

The following picture shows the internal structure of Femlisp which consists of several modules. In the figure, the module dependency is indicated by arrows. The modules appear in this form in the system definition file femlisp;femlisp.asd. They often correspond to a Common Lisp package, but may sometimes comprise several packages, where a package defines a namespace in Common Lisp.

The Femlisp modules can be roughly ordered in levels. The lowest level consists of the three modules MACROS, UTILITIES, and MATLISP. MACROS and UTILITIES extend CL with some useful macro and function definitions, while MATLISP contains a Common Lisp implementation of some BLAS and LAPACK routines which is similar to Matlisp, see Matlisp.

The second level consists of the modules ALGEBRA, MESH and GRAPHIC. GRAPHIC provides a low-level interface to external graphic software; at the moment both IBM's OpenDX and Gnuplot are supported. ALGEBRA contains the data structures and operations used for linear algebra and MESH contains mesh management including domain definitions. Both the MESH and ALGEBRA module will be discussed in more detail below.

The third level consists of the following modules:

1. The ITERATION module contains the definition for the abstract classes <solver>, <iteration>, as well as the generic function solve which constitute the interface for linear and non-linear solving. Several instances of these classes are implemented, including the conjugate gradient iteration and algebraic multigrid (AMG). The module also contains the GEOMG package, which handles iterations that depend on geometric information, e.g.~from the discretization. At the moment, these are the geometric multigrid iteration, an AMG-like scheme for preconditioning high-order discretizations with low-order ones, and some block smoothers with overlapping blocks.
2. The DISCRETIZATION module defines <discretization> as an abstract class and <fe-discretization> as a concrete derived class. A generic function get-fe is used for associating a cell with a finite element <fe>, which is a data structure containing information about base functions and dual functionals on the corresponding cell. Lagrange finite elements of arbitrary order are implemented as a special instance of <fe-discretization>. Note that other discretizations as finite differences or finite volumes could easily be incorporated as well.
3. The PROBLEM module introduces the general class <problem> and its most important subclass <pde-problem>. Several derived problems are defined, e.g. <cdr-problem> for convection-diffusion-reaction problems, <elasticity> for elasticity problems, and <navier-stokes> for Navier-Stokes problems. These problems are defined in their own packages.

The fourth level provides another level of abstraction. It consists of the modules STRATEGY and PLOT. STRATEGY provides methods for solving problems by adaptive FEM, and PLOT defines generic functions and methods for post-processing (plotting of coefficients, meshes, and functions).

The fifth level APPLICATION has access to a lot of basic modules, especially STRATEGY, DISCRETIZATION, and PLOT. There are several separate directories and files containing applications of Femlisp to special problems.

The next section describes the packages contained in these modules in detail.