Abstract
In this work we consider the computational approximation of a unique continuation problem for the Helmholtz equation using a stabilized finite element method. First conditional stability estimates are derived for which, under a convexity assumption on the geometry, the constants grow at most linearly in the wave number. Then these estimates are used to obtain error bounds for the finite element method that are explicit with respect to the wave number. Some numerical illustrations are given.
Highlights
We consider a unique continuation problem for the Helmholtz equation Δu + k2u = −f, (1)and introduce a stabilized finite element method (FEM) to solve the problem computationally
To observe optimal convergence orders of H1 and L2-errors the mesh size h must satisfy a smallness condition related to the wave number k, typically for piecewise affine elements, the condition k2h 1
This is due to the dispersion error that is most important for low order approximation spaces
Summary
We consider a unique continuation (or data assimilation) problem for the Helmholtz equation. To observe optimal convergence orders of H1 and L2-errors the mesh size h must satisfy a smallness condition related to the wave number k, typically for piecewise affine elements, the condition k2h 1. This is due to the dispersion error that is most important for low order approximation spaces. Under the assumption that the solution operator for Helmholtz problems is polynomially bounded in k, it is shown that quasi-optimality is obtained under the conditions that kh/p is sufficiently small and the polynomial degree p is at least O(log k) Another way to obtain absolute stability (i.e. stability without, or under mild, conditions on the mesh size) of the approximate scheme is to use stabilization. The above problem could arise when the acoustic wave field u is measured on ω and there are unknown scatterers present outside Ω
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