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...ing the discrete Green’s function as well as a dispersion analysis for finite element discretizations and generalizations thereof have been derived by many researchers in the past decades (see, e.g., =-=[2, 4, 6, 7, 9, 10, 11, 15, 17, 18, 19, 22, 23, 24, 25, 26, 27, 28, 33, 38, 43, 44]-=-). The goal of the present and the companion paper [32] is to derive fairly general stability and convergence estimates for Helmholtz problems that are: • explicit in the wave number, the mesh width, ...

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...rity of the pollution error in higher dimensions. One can find surveys of such methods in, e.g., [28, 20]. Broadly, they may be classified as follows: Galerkin/Least-Squares based methods [21, 29], which achieve improved stability by adding least squares residual terms to the standard Galerkin sesquilinear form; methods utilizing specialized, under-integrating quadrature rules [1] which reduce the phase error, as indicated by dispersion analysis of an interior stencil; and methods incorporating exact solutions of the Helmholtz equation (in particular, plane waves) within the trial space basis [3, 15, 16, 17, 22]. Petrov-Galerkin (PG) formulations also appear frequently in the construction of stabilized methods (see, e.g. [13, 14, 23]). Common to such methods is the introduction of local problems which are solved to provide a trial/test space pair which provides enhanced stability. A few of these methods have attempted to address in particular the Helmholtz equation. In the nearly optimal Petrov-Galerkin method (NOPG) of Barbone and Harari [5], the authors construct a method with the goal of achieving the best approximation in the H1 seminorm in a given trial space. They show that the corresponding mi...

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...mholtz type problems. However, to the best of our knowledge, none of these plane wave DG method is proved to be absolutely stable with respect to wave number k and mesh size h. We refer the reader to =-=[12, 14, 17]-=- and the references therein for more discussions in this direction. We also refer to [10, 11] for more discussions and references on other discretization techniques for the Helmholtz type problems. Th...

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...proximations instead of local absorbing boundary conditions. Similar estimates have also been investigated by Hiptmair, Moiola, and Perugia in the context of plane wave discontinuous Galerkin methods =-=[21]-=-. In section 6 we present numerical examples which support the conjecture (5.3). 5.1. Convergence of the MFS approximations. In this section we give a proof of Theorem 5.2. The proof proceeds along si...

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...ctions, and to hybridize the methods. An important paper on plane wave based discontinuous Galerkin methods (of which the UWVF is a special case, but outside the theoretical analysis) can be found in =-=[16]-=-. Coupling of plane wave and polynomial based methods is possible in the framework of that paper. Other methods that can couple plane wave and polynomial basis functions include the partition of unity...

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...ra Weak Variational Formulation (UWVF) of Cessenat and Després [6, 7, 9] belongs to the second class and permits using non-standard, discontinuous local discretization spaces such as plane waves (see =-=[16, 19]-=-). The goal of this paper is to develop a theory for the UWVF that derives the convergence behavior of abstract conforming and non-conforming generalized finite element spaces from certain local appro...

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...tion to the time-harmonic Maxwell equations, considering general Trefftz approximation spaces. Since the UWVF can be regarded as a discontinuous Galerkin (DG) method with Trefftz basis functions (see =-=[15, 22, 24]-=-), we briefly review some literature on standard (i.e., polynomial-based) DGmethods for the time-harmonic Maxwell equations. Some of them are based on the primal curl-curl formulation of the problem, ...

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...f accuracy in both the energy and the L2 norms as the standard LDG method (see [3, 12] and section 2). This work is among the series of developments in [6, 10, 11] to design discontinuous Galerkin (DG) methods with better cost efficiency for certain differential equations. Such efficiency is achieved by incorporating the a priori knowledge of the exact solutions into the choice of local approximating functions in DG formulations, and it is mainly due to the flexibility of these methods in using various local discrete spaces. Some other examples of DG methods utilizing this flexibility include [17, 8]. The objective of this note is to investigate the accuracy of the aforementioned LSP LDG method in negative-order norms. With harmonic polynomials as local approximations, the standard duality argument can not be applied to obtain the error estimates in negative-order norms for the LSP LDG method. On the other hand, negative-order norm error estimates often contain the information on the oscillatory nature of the error, which can be used to enhance the accuracy of the numerical solutions. In fact, a local post-processing technique was applied to finite element solutions of elliptic problems i...