An averaging method for the Helmholtz equation. (English) Zbl 1129.35323
Summary: The well-known Schauder result on the existence of \(\text{Lip}_\alpha (\overline{\Omega})\) solutions
of the Dirichlet problem for bounded domains with smooth boundaries is true for the
Helmholtz equation \(\Delta u+\lambda u = 0\) for \(\lambda \leq 0\). We suggest a method of constructing the
solution based on an averaging procedure and mean-value theorem. We show some
conditions under which, for \(0 < \alpha < 1\), and \(\lambda\leq 0\), a sequence of iterated averages of
an initial approximation converges geometrically to the solution.
MSC:
35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
35A35 | Theoretical approximation in context of PDEs |
65N99 | Numerical methods for partial differential equations, boundary value problems |