A Posteriori Error Estimation of $hp$-$dG$ Finite Element Methods for Highly Indefinite Helmholtz Problems

Abstract

Abstract. In this paper, we will consider an hp-finite elements discretization of a highly indefinite Helmholtz problem by some dG formulation which is based on the ultra-weak variational formulation by Cessenat and Deprés.We will introduce an a posteriori error estimator and derive reliability and efficiency estimates which are explicit with respect to the wavenumber and the discretization parameters h and p. In contrast to the conventional conforming finite element method for indefinite problems, the dG formulation is unconditionally stable and the adaptive discretization process may start from a very coarse initial mesh.Numerical experiments will illustrate the efficiency and robustness of the method.Key words. Helmholtz equation at high wavenumber, hp-finite elements, a posteriori error estimation, discontinuous Galerkin methods, ultra-weak variational formulation AMS subject classifications. 35J05, 65N12, 65N301. Introduction. High frequency scattering problems are ubiquitous in many fields of science and engineering and their reliable and efficient numerical simulation pervades numerous engineering applications such as detection (e.g., radar), communication (e.g., wireless), and medicine (e.g., sonic imaging) ([32], [1]). These phenomena are governed by systems of linear partial differential equations (PDEs); the wave equation for elastic waves and the Maxwell equations for electromagnetic scattering. We are here interested in time-harmonic problems where the equation can be reduced to purely spatial problems; for high frequencies these PDEs become highly indefinite and the development of accurate numerical solution methods is far from being in a mature state.In this paper we will consider the Helmholtz problem with high wavenumber as our model problem. Although the continuous problem with appropriate boundary conditions has a unique solution, conventional hp-finite element methods require a minimal resolution condition such that existence and uniqueness is guaranteed on the discrete level (see, e.g., [30], [29], [36], [37], [11]). However, this condition, typically, contains a generic constant C which is either unknown for specific problems or only very pessimistic estimates are available. This is one of the major motivations for the development of stabilized formulations such that the discrete system is always solvable -well-known examples include least square techniques [39,23,24,22] and discontinuous Galerkin (dG) methods [18,19,20,45,47]. These formulations lead to discrete systems which are unconditionally stable, i.e., no resolution condition is required. Although convergence starts for these methods only after a resolution condition is reached, the stability of the discrete system is considerably improved. The Ultra Weak Variational Formulation (UWVF) of Cessenat and Després [9,10,13] can be understood as a dG-method that permits the use of non-standard, discontinuous local discretization spaces such as plane waves (see [21,28,25,8]). In this paper we

For Most Frequencies, Strong Trapping Has a Weak Effect in Frequency‐Domain Scattering

“…Similar results were then obtained for DG methods in [81,101], and for least-squares methods in [10,34].…”

On the derivation of guaranteed and p-robust a posteriori error estimates for the Helmholtz equation

“…More recently, residual estimators for high-order finite element as well as discontinuous Galerkin discretizations of two-and three-dimensional problems have been studied in [28,50], where an upper bound is obtained even in the preasymptotic regime, taking the form (when the data f and g are piecewise polynomial)…”

“…One sees that in the asymptotic regime where σ ba ≤ 1, the prefactor C up in (1.3a) simplifies into an (unknown) constant that only depends on the mesh shape-regularity parameter. Moreover, the authors in [28,50] prove that as long as there are sufficiently many degrees of freedom per wavelength (essentially in the resolved regime…”

Robust Adaptive $hp$ Discontinuous Galerkin Finite Element Methods for the Helmholtz Equation

“…A DG discretization with stabilization terms also containing jumps in high order derivatives was presented in [14]. A residual-based a posteriori error estimator for the DG method of [19] is derived and analyzed in [23].…”

“…where the resolution constant C res depends on the problem under consideration; cf. [23,Section 5]. Additional numerical experiments, in which the initial conditions (5.1) are violated, are also presented in the following in order to demonstrate that the method under consideration is able to escape the pre-asymptotic region regardless of the initial mesh.…”