Interpolation properties of generalized plane waves

Abstract

This paper aims at developing new shape functions adapted to smooth vanishing coefficients for scalar wave equation. It proposes the numerical analysis of their interpolation properties. The interpolation is local but high order convergence is shown with respect to the size of the domain considered. The new basis functions are then implemented in a numerical method to solve a scalar wave equation problem with a mixed boundary condition. The order of convergence of the method varies linearly with the one of the interpolation.

Learning dominant wave directions for plane wave methods for high-frequency Helmholtz equations

“…Moreover, generalized plane waves [54][55][56] in the form e P(x) with an appropriate complex polynomial P(x) are developed to achieve high-order convergence for smooth heterogeneous media. Another instance of methods using other basis functions is the discontinuous enrichment method (DEM) [31][32][33]97], which combines Lagrange multipliers on the mesh interfaces to enforce continuity of the solution with approximation spaces composed by sums of continuous polynomials and discontinuous plane waves, leading to a reduction of the number of DOFs.…”

“…2 Adaptive methods, on the other hand, aim to leverage à priori knowledge of the solution of the Helmholtz equation, such as its known oscillatory behavior. In practice, adaptive methods have mostly focused on adaptivity to the medium, such as polynomial Galerkin methods with hp refinement [3,70,73,96,107,111], specially optimized finite differences [23,45,92,93,102] and finite elements [4,99], enriched finite elements [30][31][32][33], plane wave methods [5,21,42,43,46,69,74], generalized plane wave methods [54,55], locally corrected finite elements [17,38,82], and discretizations with specially chosen basis functions [7,8,76], among many others. They have been especially successful on reducing the pollution effect by accurately capturing the dispersion relation.…”

Extension of the nonconforming Trefftz virtual element method to the Helmholtz problem with piecewise constant wave number

“…The methods described so far have been tailored for the simplest Helmholtz problem, that is, for problems with constant wave number; the case of variable wave number is more challenging and intriguing. The instance of analytic wave number was faced in a number of works, for instance by Imbert-Gérard and collaborators in [20][21][22][23], where the so-called generalized plane waves were introduced; the idea behind that approach is to employ approximation spaces that are globally discontinuous and locally spanned by combinations of exponential functions applied to complex polynomials. It is worthwhile to notice that this method is quasi-Trefftz only (that is, when applying the Helmholtz operator to the basis functions, one gets a quantity which is converging to zero as the mesh size decreases and the dimension of the local space increases) and that it generalizes the discontinuous enrichment method [31], which addresses the simpler case of linear wave number.…”

Trefftz approximations in complex media: Accuracy and applications

“…One can refer, for example, to papers by Melenk, Hiptmair, Moiola, Perugia et al cited above, to the references in these papers, and to Perrey-Debain's paper [24]. Much less attention has been paid to the inhomogeneous case [25, Chapter IV], [26,Section 3], [27,28], which is substantially more complicated but at the same time more rewarding in practice.…”