A Plane Wave Virtual Element Method for the Helmholtz Problem

We introduce and analyze a virtual element method (VEM) for the Helmholtz problem with approximating spaces made of products of low order VEM functions and plane waves. We restrict ourselves to the 2D Helmholtz equation with impedance boundary conditions on the whole domain boundary. The main ingredients of the plane wave VEM scheme are: i) a low frequency space made of VEM functions, whose basis functions are not explicitly computed in the element interiors; ii) a proper local projection operator onto the high-frequency space, made of plane waves; iii) an approximate stabilization term. A convergence result for the h-version of the method is proved, and numerical results testing its performance on general polygonal meshes are presented

A survey of Trefftz methods for the Helmholtz equation

Trefftz methods are finite element-type schemes whose test and trial functions are (locally) solutions of the targeted differential equation. They are particularly popular for time-harmonic wave problems, as their trial spaces contain oscillating basis functions and may achieve better approximation properties than classical piecewise-polynomial spaces. We review the construction and properties of several Trefftz variational formulations developed for the Helmholtz equation, including least squares, discontinuous Galerkin, ultra weak variational formulation, variational theory of complex rays and wave based methods. The most common discrete Trefftz spaces used for this equation employ generalised harmonic polynomials (circular and spherical waves), plane and evanescent waves, fundamental solutions and multipoles as basis functions; we describe theoretical and computational aspects of these spaces, focusing in particular on their approximation properties. One of the most promising, but not yet well developed, features of Trefftz methods is the use of adaptivity in the choice of the propagation directions for the basis functions. The main difficulties encountered in the implementation are the assembly and the ill-conditioning of linear systems, we briefly survey some strategies that have been proposed to cope with these problems.Comment: 41 pages, 2 figures, to appear as a chapter in Springer Lecture Notes in Computational Science and Engineering. Differences from v1: added a few sentences in Sections 2.1, 2.2.2 and 2.3.1; inserted small correction

Mixed finite elements for numerical weather prediction

We show how two-dimensional mixed finite element methods that satisfy the conditions of finite element exterior calculus can be used for the horizontal discretisation of dynamical cores for numerical weather prediction on pseudo-uniform grids. This family of mixed finite element methods can be thought of in the numerical weather prediction context as a generalisation of the popular polygonal C-grid finite difference methods. There are a few major advantages: the mixed finite element methods do not require an orthogonal grid, and they allow a degree of flexibility that can be exploited to ensure an appropriate ratio between the velocity and pressure degrees of freedom so as to avoid spurious mode branches in the numerical dispersion relation. These methods preserve several properties of the C-grid method when applied to linear barotropic wave propagation, namely: a) energy conservation, b) mass conservation, c) no spurious pressure modes, and d) steady geostrophic modes on the $f$-plane. We explain how these properties are preserved, and describe two examples that can be used on pseudo-uniform grids: the recently-developed modified RT0-Q0 element pair on quadrilaterals and the BDFM1-\pdg element pair on triangles. All of these mixed finite element methods have an exact 2:1 ratio of velocity degrees of freedom to pressure degrees of freedom. Finally we illustrate the properties with some numerical examples.Comment: Revision after referee comment

On stability of discretizations of the Helmholtz equation (extended version)

We review the stability properties of several discretizations of the Helmholtz equation at large wavenumbers. For a model problem in a polygon, a complete $k$-explicit stability (including $k$-explicit stability of the continuous problem) and convergence theory for high order finite element methods is developed. In particular, quasi-optimality is shown for a fixed number of degrees of freedom per wavelength if the mesh size $h$ and the approximation order $p$ are selected such that $kh/p$ is sufficiently small and $p = O(\log k)$, and, additionally, appropriate mesh refinement is used near the vertices. We also review the stability properties of two classes of numerical schemes that use piecewise solutions of the homogeneous Helmholtz equation, namely, Least Squares methods and Discontinuous Galerkin (DG) methods. The latter includes the Ultra Weak Variational Formulation

Hybrid quadrilateral finite element models for axial symmetric Helmholtz problem

This paper is a continuation of the previous work in which six-node triangular finite element models for the axial symmetric Helmholtz problem are devised by using a hybrid functional and the spherical-wave modes [1]. The six-node models can readily be incorporated into the standard finite element program framework and are typically ∼50% less erroneous than their conventional or, equivalently, continuous Galerkin counterpart. In this paper, four-node and eight-node quadrilateral models are devised. Two ways of selecting the spherical-wave modes are attempted. In the first way, a spherical-wave pole is selected such that it is equal-distant from an opposing pair of element nodes. In the second way, the directions of the spherical-waves passing through the element origin are equal-spaced with one of the directions bisecting the two parametric axes of the element. Examples show that both ways lead to elements that yield very similar predictions. Furthermore, four-node and eight-node hybrid elements are typically ∼50% and ∼70% less erroneous than their conventional counterparts, respectively. © 2011 Elsevier B.V. All rights reserved.postprin