126 research outputs found
Stabilized Galerkin for transient advection of differential forms
No full textWe deal with the discretization of generalized transient advection problems for differential forms on bounded spatial domains. We pursue an Eulerian method of lines approach with explicit timestepping. Concerning spatial discretization we extend the jump stabilized Galerkin discretization proposed in [H. Heumann and R. Hiptmair, Stabilized Galerkin methods for magnetic advection, Math. Modelling Numer. Analysis, 47 (2013), pp. 1713{ 1732] to forms of any degree and, in particular, advection velocities that may have discontinuities resolved by the mesh. A rigorous a priori convergence theory is established for Lipschitz continuous velocities, conforming meshes and standard nite element spaces of discrete differential forms. However, numerical experiments furnish evidence of the good performance of the new method also in the presence of jumps of the advection velocity
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Trefftz discontinuous Galerkin methods for acoustic scattering on locally refined meshes
No full textWe extend the a priori error analysis of Trefftz-discontinuous Galerkin methods for time-harmonic wave propagation problems developed in previous papers to acoustic scattering problems and locally refined meshes. To this aim, we prove refined regularity and stability results with explicit dependence of the stability constant on the wave number for non convex domains with non connected boundaries. Moreover, we devise a new choice of numerical flux parameters for which we can prove L2-error estimates in the case of locally refined meshes near the scatterer. This is the setting needed to develop a complete hp-convergence analysis
Integral Equations for Acoustic Scattering by Partially Impenetrable Composite Objects
No full textInternational audienceWe study direct first-kind boundary integral equations arising from transmission problems for the Helmholtz equation with piecewise constant coefficients and Dirichlet boundary conditions imposed on a closed surface. We identify necessary and sufficient conditions for the occurrence of so-called spurious resonances, that is, the failure of the boundary integral equations to possess unique solutions.Following [A. Buffa and R. Hiptmair, Numer Math, 100, 1–19 (2005)] we propose a modified version of the boundary integral equations that is immune to spurious resonances. Via a gap construction it will serve as the basis for a universally well-posed stabilized global multi-trace formulation that generalizes the method of [X. Claeys and R. Hiptmair, Commun Pure and Appl Math, 66, 1163–1201 (2013)] to situations with Dirichlet boundary conditions
SPURIOUS QUASI-RESONANCES IN BOUNDARY INTEGRAL EQUATIONS FOR THE HELMHOLTZ TRANSMISSION PROBLEM
No full textWe consider the Helmholtz transmission problem with piecewise-constant material coefficients and the standard associated direct boundary integral equations. For certain coefficients and geometries, the norms of the inverses of the boundary integral operators grow rapidly through an increasing sequence of frequencies, even though this is not the case for the solution operator of the transmission problem; we call this phenomenon that of spurious quasi-resonances. We give a rigorous explanation of why and when spurious quasi-resonances occur and propose modified boundary integral equations that are not affected by the
Substructuring the Hiptmair-Xu preconditioner for positive Maxwell problems
No full textConsidering positive Maxwell problems, we propose a substructured version of
the Hiptmair-Xu preconditioner based on a new formula that expresses the
inverse of Schur systems in terms of the inverse matrix of the global volume
problem
Trefftz Co-chain Calculus
No full textWe propose a comprehensive approach to obtain systems of equations that discretize linear stationary or time-harmonic elliptic problems in unbounded domains. This is achieved by coupling any numerical method that fits co-chain calculus with a Trefftz method. The framework of co-chain calculus accommodates both finite element exterior calculus and discrete exterior calculus. It encompasses methods based on volume meshes: its application is therefore confined to bounded domains.
Conversely, Trefftz methods are based on functions that solve the homogeneous equations exactly in the unbounded complement of the meshed domain, while satisfying suitable conditions at infinity. An example of a Trefftz method is the Multiple Multipole Program (MMP), which makes use of multipoles, i.e. solutions spawned by point sources with central singularities that are placed outside the domain of approximation. In our approach the degrees of freedom describing these sources can be eliminated by computing the Schur complement of the system for the coupling, therefore leading to a boundary term for co-chain calculus that takes into account the exterior problem. As a concrete example, we specialize this general framework for the cell method, a particular variant of discrete exterior calculus, coupled with MMP to solve frequency-domain eddy-current problems. A numerical experiment shows the effectiveness of this approach
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Plane wave approximation of homogeneous Helmholtz solutions
No full textIn this paper, we study the approximation of solutions of the homogeneous Helmholtz equation Δu + ω 2 u = 0 by linear combinations of plane waves with different directions. We combine approximation estimates for homogeneous Helmholtz solutions by generalized harmonic polynomials, obtained from Vekua’s theory, with estimates for the approximation of generalized harmonic polynomials by plane waves. The latter is the focus of this paper. We establish best approximation error estimates in Sobolev norms, which are explicit in terms of the degree of the generalized polynomial to be approximated, the domain size, and the number of plane waves used in the approximations
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Vekua theory for the Helmholtz operator
No full textVekua operators map harmonic functions defined on domain in \mathbb R2R2 to solutions of elliptic partial differential equations on the same domain and vice versa. In this paper, following the original work of I. Vekua (Ilja Vekua (1907–1977), Soviet-Georgian mathematician), we define Vekua operators in the case of the Helmholtz equation in a completely explicit fashion, in any space dimension N ≥ 2. We prove (i) that they actually transform harmonic functions and Helmholtz solutions into each other; (ii) that they are inverse to each other; and (iii) that they are continuous in any Sobolev norm in star-shaped Lipschitz domains. Finally, we define and compute the generalized harmonic polynomials as the Vekua transforms of harmonic polynomials. These results are instrumental in proving approximation estimates for solutions of the Helmholtz equation in spaces of circular, spherical, and plane waves
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