On the Stability of Piecewise Linear Wavelet Collocation and the Solution of the Double Layer Equation over Polygonal Curves
- Rathsfeld, Andreas
2010 Mathematics Subject Classification
- 45L10 65R20 65N38
- collocation, wavelet algorithm, double layer potential
In this paper we consider a piecewise linear collocation method for the solution of strongly elliptic operator equations over closed curves. The trial space is a subspace of the space of all piecewise linear functions defined over a uniform grid. This space is spanned by an arbitrary subset of the biorthogonal wavelet basis. To the subspace in the trial space there corresponds a natural subspace in the space of test functionals. This subspace is spanned by certain linear combinations of the Dirac delta functionals taken at the uniformly distributed grid points. For the resulting wavelet collocation method and a strongly elliptic operator equation, we prove stability and convergence. In particular, this general result applies to the double layer equation over a polygonal curve. We show that the wavelet collocation method with piecewise linear trial functions over a uniform grid converges with order O(n-2), where n is the number of degrees of freedom. Note that the step size of the underlying uniform partition is n-ɑ, ɑ ≥ 1. The stiffness matrix for the wavelet collocation method can be compressed to a matrix containing no more than O(n log n) non-zero entries such that the asymptotic convergence order is not effected.
- Comput. Eng., 1, ed. M. Goldberg, WIT Press/Comput. Mech. Publ., Boston, MA, 1999