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Non-uniform finite-element meshes defined by ray dynamics for Helmholtz problems
Martin Averseng, Jeffrey Galkowski, Euan A. Spence
The -version of the finite-element method (-FEM) applied to the high-frequency Helmholtz equation has been a classic topic in numerical analysis since the 1990s. It is now ri…
Boundary Element Methods for the Laplace Hypersingular Integral Equation on Multiscreens: a two-level Substructuring Preconditioner
Martin Averseng, Xavier Claeys, Ralf Hiptmair
We present a preconditioning method for the linear systems arising from the boundary element discretization of the Laplace hypersingular equation on a -dimensional triangulated…
Helmholtz FEM solutions are locally quasi-optimal modulo low frequencies
Martin Averseng, Euan A. Spence, Jeffrey Galkowski
For -FEM discretisations of the Helmholtz equation with wavenumber , we obtain -explicit analogues of the classic local FEM error bounds of [Nitsche, Schatz 1974], [Wahlbi…
Fractured Meshes
Martin Averseng, Xavier Claeys, Ralf Hiptmair
This work introduces ``generalized meshes", a type of meshes suited for the discretization of partial differential equations in non-regular geometries. Generalized meshes extend re…
Jump-preserving polynomial interpolation in non-manifold polyhedra
Martin Averseng
We construct a piecewise-polynomial interpolant for functions , where is a Lipschitz polyhedron and $Γ\subset Ω…
Pseudo-differential analysis of the Helmholtz layer potentials on open curves
Martin Averseng
We introduce two new classes of pseudo-differential operators on open curves. They correspond via a change of variables to subclasses of the periodic pseudo-differential operators,…
New preconditioners for Laplace and Helmholtz integral equations on open curves: Analytical framework and Numerical results
François Alouges, Martin Averseng
The Helmholtz wave scattering problem by screens in 2D can be recast into first-kind integral equations which lead to ill-conditioned linear systems after discretization. We introd…
Fast discrete convolution in using Sparse Bessel Decomposition
Martin Averseng
We describe an efficient algorithm for computing the matrix vector products that appear in the numerical resolution of boundary integral equations in 2 space dimension. This work i…