Research
My main interest is the analysis of boundary integral equations on non smooth surfaces and the development of numerical methods for their approximate solution. The usual examples of non smooth surfaces are polygonal surfaces. Here the surface is the union of lines (in two dimensions) or triangles (in three dimensions). These kind of surfaces are used very often in CAD systems to model realistic surfaces.
The solutions of boundary integral equations on these non smooth surfaces are not smooth along the intersection of two triangles (in three dimensions) or at the intersection of two boundary lines (in two dimensions). Therefore one has to use special meshes for the numerical approximation in order to get the optimal convergence rates, for example quadratic convergence rates if one uses linear trial functions. Beside this the integral equations on non smooth surfaces are often not of the type identity I+K, where K is a compact operator. This implies that one cannot apply the standard stability theory for the numerical solution of Fredholm integral equations of the second kind.
I studied this kind of problems in different situations. One application was the Lame equation in two dimensions with Dirichlet boundary conditions. Here one gets a system of integral equations on the boundary. Together with Dr. J. Elschner I was able to derive a stability result for the collocation method for this equation. Then I studied time dependent problems for the heat equation. I investigated a single and a double layer ansatz for this problem and a collocation method for the numerical solution. Both methods worked well for polygonal boundaries. Especially one gets the optimal order of convergence on polygonal boundaries if the right kind of graded meshes is used. For the double layer potential I was able to prove a stability result.
In 2000 I started to study the radiosity equation. This equation describes the exchange of energy between different parts of a surface by radiation. Prof. K. Atkinson, University of Iowa, Iowa City, proposed this subject to me and he already got various results on the stability of the collocation method for the radiosity equation and the influence of shadows on the performance of numerical methods. I was able to derive some properties of the solution of the radiosity equation in the general three dimensional case. Furthermore I was proved a stability result for the collocation method in the 3D case if one uses graded meshes and modification of the i* trick, invented by G. Chandler and I. Graham for the two dimensional collocation method. I started to investigate hierarchical methods for the solution of the radiosity equation. These kind of methods are very common in practical calculations. My first results cover only the two dimensional case and surfaces without any shadow. But the analysis shows that these methods seem to be very efficient.
In 2003 I worked also on an inverse problem for integral equations where the kernel of the integral equation has a Mellin structure. Together with R. Ramlau and S. Fischer, University of Bremen, I was able to derive a numerical algorithm and S. Fischer implemented the method. The results show that the algorithm works in practice. I also started to analyze a spherical harmonics extension model for charged particles attracted to a plate by some potential. This model was derived by P. Degond. Together with A. Juengel, University of Mainz, I was able to prove a long time existence result and the conservation of mass for this equation.
Since 2005 I am working on methods for the solution of boundary integral equations which use polynomial trial functions. These methods guarantee fast convergence if the solution is a smooth function. Therefore these methods are well suited for smooth domains like disks or spheres.