Title  Sparsegrid and prewavelet based methods for the solution of integral equations  
Participiants  Thomas Schiekofer Michael Griebel  
Keywords  Integral equations, Riesz basis, Sparse Grids  
Description 
We are interested in the solution of boundary integral equations on a
twodimensional manifold in
(screen problems). Here, we deal with the
single layer potential equation
which we will write as operator equation Vf = g where V is a pseudodifferential operator of order 1. This problem is equivalent to the Dirichlet problem for the Laplace equation in the exterior domain with Dirichlet data g on . We are furthermore interested in the hypersingular equation
which we will write as operator equation Df = g where D is a pseudodifferential operator of order 1, and in the double layer potential equation
which corresponds to the interior Neumann problem. We solve the integral equations with the BEM where we use semiorthogonal Riesz bases as ansatz functions as well as test functions (Galerkin method). In the numerical tests we use two different examples for such Riesz bases:
In the following, Figure 1 shows the generating functions for the piecewise constant Riesz bases (Figure 1 a) as well as for the semiorthogonal linear spline wavelets (Figure 1 b,c). The grid point sets of both above examples of Riesz bases can be seen in Figure 2. Here, the onedimensional case is shown in the top of the figure. In the middle of this figure the full grid case can be seen, while the sparse grid case is presented at the bottom.
The discretization of each of the above integral equations leads to a linear system of equations where the entries of the stiffness matrices can be computed exactly using recurrence formulae which can be found in [8]. Obviously, the resulting stiffnes matrices are symmetric positive definite as we use the same bases for the ansatz and test functions. Hence, we use the conjugate gradients method as an iterative solver where we use Jacobian preconditioning. We have run comparative test for both above mentioned Riesz bases. All tests are for the capacity problem for a unit square screen with
for the single layer potential equation with right hand side g=1, e.g. we compute approximate solutions of the problem Vf=1. The capacity of a screen is defined as the average of the solution f of the equation Vf=1:
Let us denote by L the Level of the discretization (of a full grid and a a sparse grid, respectively), and let and be the Galerkin solutions of the single layer potential equation for the full grid space and the sparse grid space , respectively. It is wellknown that
holds. Therefore, it is enough to concentrate on capacity errors. In the following, some numerical examples are shown. In the tables we use the abbrevations L for Level, K for kondition number of the stiffness matrix, It for the number of iterations to reduce the relative error by 10^{6} in the residual of the preconditioned system, E(C) for the capacity error and q for the quotient in the capacity error showing the numerical convergence of the scheme. First, the results obtained using the Haar basis are presented.
The results of Table 2 can also be seen in the following two figures (Figure 3 and Figure 4) where the capacity error and the condition number vs. degrees of freedom can be seen. Here, we are almost in the asymptotic range, and we already see that the condition number is bounded from above as theory predicts.
 
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