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

Abstract

We extend the nonconforming Trefftz virtual element method introduced in [30] to the case of the fluid-fluid interface problem, that is, a Helmholtz problem with piecewise constant wave number. With respect to the original approach, we address two additional issues: firstly, we define the coupling of local approximation spaces with piecewise constant wave numbers; secondly, we enrich such local spaces with special functions capturing the physical behavior of the solution to the target problem. As these two issues are directly related to an increase of the number of degrees of freedom, we use a reduction strategy inspired by [31], which allows to mitigate the growth of the dimension of the approximation space when considering h- and p-refinements. This renders the new method highly competitive in comparison to other Trefftz and quasi-Trefftz technologies tailored for the Helmholtz problem with piecewise constant wave number. A wide range of numerical experiments, including the p-version with quasi-uniform meshes and the hp-version with isotropic and anisotropic mesh refinements, is presented.

Introduction

Efficient methods for the approximation of solutions to high frequency wave propagation problems have received an increasing attention over the last two decades. Starting from the ultra weak variational formulation of Cessenat and Després [12], many wave based methods for the Helmholtz problem have been introduced and analyzed, see [21] for an overview of the topic. Such methods are in general based on trial and test spaces consisting of piecewise (discontinuous) plane waves.

In the framework of the virtual element method (VEM) [5], [6], which can be seen as an extension of the finite element method (FEM) to polytopal meshes and as the ultimate evolution of the mimetic finite differences [8], [26], an $H^1$-conforming method for the Helmholtz problem was introduced in [32]. Such a method, known as the plane wave VEM, is based on local approximation spaces containing plane waves that are eventually patched continuously with the aid of a partition of unity, in the spirit of the pioneering work of Melenk and Babuška [4].

More recently, a novel nonconforming Trefftz-VEM for the Helmholtz problem was developed in [30] as an extension of the harmonic VEM [13], [29] for the Laplace problem. The two main features of this method are (i) that it is Trefftz (i.e., local spaces consist of functions belonging to the kernel of the target differential operator) and $(ii)$ that it falls within the nonconforming virtual element framework, see e.g. [2], [3], [11], [17]. Although in the basic construction of the method more degrees of freedom than e.g. in the plane wave discontinuous Galerkin method [19] are needed, a modification of a strategy introduced in [31] allows to significantly reduce the dimension of the approximation space as well as the condition number of the resulting final system; this renders the nonconforming Trefftz-VEM approach highly competitive in comparison with other Trefftz technologies. Roughly speaking, the main idea of this strategy is that, whenever two basis functions are generating “almost” the same space, one of the two can be kicked out from the set of basis functions, yet not jeopardizing the approximation properties of the space.

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 [16], [23], [24], [25], 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 [33], which addresses the simpler case of linear wave number. Another quasi-Trefftz method for smooth wave numbers is provided in a work of Betcke and Phillips in [10]; there, the basis functions are modulated plane waves, i.e., products of plane waves with polynomials.

On the other hand, the instance of piecewise constant wave numbers gives raise to the fluid-fluid interface problem, which models the transmission of a wave between two fluids with different refraction indices; such model is in fact the one tackled in the present paper. We mention that the plane wave discontinuous Galerkin method and the discontinuous enrichment method have been successfully applied to this problem, see [27] and [34], respectively. In those two approaches, Bessel functions were employed in addition to plane waves, and other special functions (namely evanescent waves) were added to capture the physical behavior of the solution at the interface between the two fluids.

In this paper,

• we extend the nonconforming Trefftz-VEM of [30], [31] to the case of piecewise constant wave numbers, and

• following what was done in [27], [34], we also include proper special functions in the approximation spaces to capture the behavior of the physical solution.

We will see that both issues elegantly fit within the nonconforming Trefftz-VEM framework. Further, by employing the removing technique introduced in [31], an extremely robust numerical performance is obtained.

The method we are going to present is characterized by local spaces containing plane (and possibly evanescent) waves, plus additional functions implicitly defined as solutions to local Helmholtz problems with impedance boundary conditions in proper 1D plane and evanescent wave spaces. These local spaces are eventually coupled in a nonconforming fashion à la Crouzeix-Raviart (in the sense that the jumps across the interface between elements have zero moments up to a certain order). The fact that the functions in the approximation space are unknown in closed form entails that, in order to implement the method, one can not use the continuous sesquilinear form; rather, discrete counterparts based on projections onto (plane and evanescent) wave spaces and stabilizing sesquilinear forms are employed.

The outline of the paper is as follows. Section 2 is devoted to the description of the model problem, whereas Section 3 provides the notation for plane wave and evanescent wave spaces, as well as for nonconforming Sobolev spaces. The method, including the definition of the local and the global spaces, of a set of degrees of freedom, of suitable projections onto wave spaces, and of suitable stabilizations, is the topic of Section 4. In Section 5, we briefly discuss the implementation details of the method and we present a number of numerical experiments. In particular, we study the performance of the h- and of p-versions, whenever the meshes are conforming with respect to the interface between the two fluids (i.e., the wave number is piecewise constant over the polygonal decomposition); the rate of convergence is algebraic and exponential in terms of h and p in the former and in the latter case, respectively. Another interesting set of experiments is focused on testing the robustness of the method, whenever some elements are cut by the interface; on such elements, in fact, the solution to the fluid-fluid problem has typically a very low Sobolev regularity, and therefore the convergence of the h- and of p-versions is poor. Consequently, the hp-version with geometric isotropic and anisotropic mesh refinements is employed, leading to algebraic and exponential convergence in terms of proper roots of the number of degrees of freedom in the former and in the latter case, respectively. Some conclusions are stated in Section 6. It is important to highlight that in the implementation of the method, quadrature formulas are needed only for the approximation of the terms involving the boundary data.

We stress that, although the present paper is aimed at the approximation of the Helmholtz problem with piecewise constant wave number solely, the setting of the nonconforming Trefftz-VEM can be applied in other situations. For instance, one could extend the method to the case of analytic wave number, dovetailing the nonconforming VEM technology with the tools stemming from the theory of generalized plane waves. A possible advantage of employing a variant of the approach presented herein, in lieu of the discontinuous Galerkin one [25], is that the orthogonalization-and-filtering technique inspired by [31] could lead to an improved convergence rate in terms of the number of degrees of freedom and to an improved conditioning of the final system.

As a final comment, we stress that another appealing feature of the nonconforming setting is that the extension to the 3D case is much more straightforward than in the $H^1$-conforming setting; see [29, Section 3.7] for a description of such an extension in the case where the target differential operator is the Laplacian.

Notation.  Throughout the paper, we will employ the standard notation for Sobolev spaces, norms, seminorms and inner products, see e.g. [1]. More precisely, given a domain $D\subset {\mathbb{R}}^2$, we denote by $H^s(D)$ the Sobolev space of functions with square integrable weak derivatives up to order s, for some nonnegative integer s, over D, and the corresponding seminorms and norms by $|\cdot {|}_{s,D}$ and ${\Vert \cdot \Vert }_{s,D}$, respectively. Sobolev spaces of noninteger order can be defined by interpolation theory. If the domain D is also bounded, $H^{1/2}(\partial D)$ denotes the space of the traces of $H^1(D)$ functions and $H^{-1/2}(\partial D)$ denotes its dual space. Further, ${(\cdot ,\cdot )}_{0,D}$ is the usual $L^2$ inner product over D. Lastly, we denote by ${\mathbb{N}}_0$ the set of all natural numbers including 0, and by ${\mathbb{N}}_{\ge r}$, for some $r>0$, the set of all natural numbers larger than or equal to r.