Solving Poisson’s equation on a general shape using finite differences

One of the questions I received in the comments to my old post on solving Laplace equation (in fact this is Poisson’s equation) using finite differences was how to apply this procedure on arbitrary domains. It is not possible to use this method directly on general domains. The main problem is the fact that, unlike in the case of a square of rectangular domain, when we have a general shape, the boudary can have any orientation, not only the orientation of the coordinate axes. One way to avoid approach this problem would be using the Finite Element Method. Briefly, you discretize the boundary, you consider a triangulation of the domain with respect to this discretization, then you consider functions which are polynomial and have support in a few number of triangles. Thus the problem is reduced to a finite dimensional one, which can be written as a matrix problem. The implementation is not straightforward, since you need to conceive algorithms for doing the discretization and triangulation of your domain.

One other approach is to consider a rectangle which contains the shape and add a penalization on the exterior of your domain . The problem to solve becomes something like:

where is defined by

Note that doing this we do not need to impose the boundary condition on . This is already imposed by , and the fact that is forced to be zero outside .

Eigenvalues via Fundamental Solutions

Eigenvalue problems like

can be solved numerically in a variety of ways. Probably the best known one is the finite element method. I will present below the sketch of an algorithm which does not need meshes, and when implemented correctly, can decrease computational costs.

The idea of Fundamental Solution first appeared in the 60s and was initially used to find solutions of the Laplace equation in a domain. It later was extended to more general equations and eigenvalue problems. The method uses (as the title says) some particular fundamental solutions of the studied equation to create an approximation of the solution as a linear combination of them. The advantage is that the fundamental solutions are sometimes known in analytic form, and the only thing that remains to do is to find the optimal coefficients in the linear combination. A detailed exposure of the method can be found in the following article of Alvez and Antunes.

First Dirichlet eigenvalue is simple for connected domains

Suppose is a connected open set and consider the first two eigenvalues of the Laplace operator with Dirichlet boundary conditions . Then .

Numerical Results – Optimal Shapes – Dirichlet Eigenvalues – Perimeter Constraint

Recently it was proved that the problem

has a solution even if is not confined to a bounded open set . ( is the -th eigenvalue for the Laplace operator with Dirichlet boundary conditions.)

A similar result was proven recently by B. Velichkov and G. de Philippis for the case where the volume constraint is replaced by a perimeter constraint, i.e. the problem

has a solution, and this solution is regular enough.

Numerical Results – Optimal Shapes – Dirichlet Eigenvalues – Volume Constraint

The only known exact shapes which minimize the eigenvalues of the Laplace operator with Dirichlet condition are those for and . Nothing is proved for higher eigenvalues, but there are some numerical tests which were performed to find what the optimal shapes look like. Such tests were made first by Edouard Oudet for and recently by P. Freitas, P. Antunes for .

Weak formulation for Laplace Equation with Robin boundary conditions

Consider an open set with Lipschitz boundary and consider on the following problem

where is a constant. This is the Laplace equation with Robin boundary conditions. I will prove that the problem is well posed and for each there exists a solution .