Abstract
Helmholtz problem appears in many areas for example in the context of inverse and scattering problems. This problem is solved numerically and the challenge is that the solutions become highly oscillatory. As a consequence the numerical discretization has to be adapted to resolve these oscillations. Galerkin methods are well established to solve elliptic problems - however for Helmholtz problems they suffer from the indefiniteness of the equation, more precisely, the stability of the discrete solution as well as the corresponding error is significantly “polluted” in the preasymptotic range. For high wave numbers, the solution shows a non-robust behavior which is known as the pollution effect, i.e. the discrepancy between the best approximation error and the error of the Galerkin solution increases with increasing wave number. The physical reason for this is the highly oscillatory nature of the solution of this problem while in the mathematical language the Helmholtz equation becomes highly indefinite with increasing wave number. It is an important topic of research in numerical analysis to find an efficient numerical discretization which behaves reasonably robust with respect to the wave number. One of the interesting questions in this area is how the performance of the method can be affected by the parameters like the wave number and the mesh size. Classical conforming low-order finite elements suffer from pollution effect. The minimal dimension, N, e.g., of the P1-finite elements must satisfy N & k2d where d is spatial dimension. Previous works show that the high oscillation of the solution can be resolved by refining the mesh size, h, and the pollution effect can be reduced by employing higher order methods with comparison to lower order methods. For example it is known that for conforming finite elements we can get a more relaxed condition (N & kd) if we use higher order methods. In the recent years much progress has been made for nonconforming finite element discretizations of the Helmholtz problem. Among them are the plane wave discretization in combination with the ultra-weak variational formulation
which turn out to be unconditionally stable. In this thesis, we develop a stability and convergence theory for the Ultra Weak Variational Formulation (UWVF) of a highly indefinite Helmholtz problem in Rd,d = 1,2,3 for general, abstract trial and test spaces. The theory covers conforming as
ii well as nonconforming generalized finite element methods. In contrast to conventional Galerkin methods where a minimal resolution condition is necessary to guarantee the unique solvability, it is proved that the UWVF admits a unique solution without any condition for piecewise polynomials and plane waves and under a mild condition for a very general class of the approximation spaces. We develop a theory for general abstract non-conforming Galerkin discretization. As an application we present the error analysis for the conforming and non-conforming hp-version of the finite element method explicitly in terms of the mesh width h, polynomial degree p and wave number k for two different cases. We show that our method converges with almost optimal order under the conditions that kh/p is sufficiently small and the polynomial degree p is at least O(logk).
Asieh, P