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We present a first order system least squares (FOSLS) method for the Helmholtz equation at high wave number k, which always leads to a Hermitian positive definite algebraic system. By utilizing a non-trivial solution decomposition to the dual FOSLS problem which is quite different from that of the standard finite element methods, we give an error analysis to the hp-version of the FOSLS method where the dependence on the mesh size h, the approximation order p, and the wave number k is given explicitly. In particular, under some assumption of the boundary of the domain, the L 2 norm error estimate of the scalar solution from the FOSLS method is shown to be quasi optimal under the condition that kh/p is sufficiently small and the polynomial degree p is at least O(log k). Numerical experiments are given to verify the theoretical results.2000 Mathematics Subject Classification. 65N30, 65L12.

The concept of the M -decomposition was introduced by Cockburn et al. in Math. Comp. vol. 86 (2017), pp. 1609-1641 to provide criteria to guarantee optimal convergence rates for the Hybridizable Discontinuous Galerkin (HDG) method for coercive elliptic problems. In that paper they systematically constructed superconvergent hybridizable discontinuous Galerkin (HDG) methods to approximate the solutions of elliptic PDEs on unstructured meshes. In this paper, we use the M -decomposition to construct HDG methods for the Maxwell's equations on unstructured meshes in two dimension. In particular, we show the any choice of spaces having an M -decomposition, together with sufficiently rich auxiliary spaces, has an optimal error estimate and superconvergence even though the problem is not in general coercive. Unlike the elliptic case, we obtain a superconvergent rate for the curl of the solution, not the solution, and this is confirmed by our numerical experiments. of problems posed in two dimensions. To see how this is possible, consider the usual time harmonic Maxwell system for the electric field E (a complex valued vector function):

Abstract. A local multilevel product algorithm and its additive version are analyzed for linear systems arising from the application of adaptive finite element methods to second order elliptic boundary value problems. The abstract Schwarz theory is applied to verify uniform convergence of local multilevel methods featuring Jacobi and Gauss-Seidel smoothing only on local nodes. By this abstract theory, convergence estimates can be further derived for the hierarchical basis multigrid method and the hierarchical basis preconditioning method on locally refined meshes, where local smoothing is performed only on new nodes. Numerical experiments confirm the optimality of the suggested algorithms.1. Introduction. Multigrid or multilevel methods belong to the most efficient methods to solve large linear systems arising from the discretization of elliptic boundary value problems by finite element methods. The convergence properties of multigrid methods for conforming finite elements have been studied by many authors (cf., e.g., [9] In particular, using the notions of space decomposition and subspace correction, a unified theory has been established in [32] for a general class of iterative algorithms such as multigrid methods, overlapping domain decomposition methods, and hierarchical basis methods.In this paper, we study local multilevel methods for adaptive finite element methods (AFEM) applied to second order elliptic boundary value problems. Mesh adaptivity based on a posteriori error estimators has become a powerful tool for solving partial differential equations. It is known that the convergence property of AFEM with the newest vertex bisection algorithm is optimal in the sense that the finite element discretization error is proportional to N −1/2 in the energy norm, where N is the number of degrees of freedom on the underlying mesh (cf., e.g., [7] [26]. We emphasize that these locally refined meshes obey restrictive conditions which are not satisfied by the newest vertex bisection algorithm which will be used for adaptivity in this work. As far as AFEM procedures featuring the newest vertex bisection algorithm are concerned, Wu and Chen [30] have been the first to show that the multigrid V-cycle algorithm performng Gauss-Seidel smoothing on new nodes and those old nodes where the support of the associated nodal basis function has changed can guarantee uniform convergence of the algorithm.The objective of this paper is to utilize the well-known Schwarz theory [29] to study local multilevel methods with local Jacobi or local Gauss-Seidel smoothing. Within this framework