FINITE ELEMENT SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATION WITH MULTIPLE CONCAVE CORNERS

In [8] they introduced a new _nite element method for accurate numerical solutions of Poisson equations with corner sin- gularities. They consider the Poisson equations with homogeneous Dirichlet boundary condition with one corner singularity at the ori- gin, and compute the _nite element solution using standard FEM and use the extraction formula to compute the stress intensity fac- tor, then pose a PDE with a regular solution by imposing the non- homogeneous boundary condition using the computed stress inten- sity factor, which converges with optimal speed. From the solution they could get an accurate solution just by adding the singular part. This approach uses the polar coordinate and the cut-off function to control the singularity and the boundary condition. In this paper we consider Poisson equations with multiple sin- gular points, which involves different cut-off functions which might overlaps together and shows the way of cording in FreeFEM++ to control the singular functions and cut-off functions with numerical experiments.