A first order system least squares method for the Helmholtz equation

We present a first order system least squares (FOSLS) method for the Helmholtz equation at high wave number k, which always deduces Hermitian positive definite algebraic system. By utilizing a non-trivial solution decomposition to the dual FOSLS problem which is quite different from that of standard finite element method, we give 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 L2 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

Boundary conditions for the numerical solution of elliptic equations in exterior regions

Elliptic equations in exterior regions frequently require a boundary condition at infinity to ensure the well-posedness of the problem. Examples of practical applications include the Helmholtz equation and Laplace's equation. Computational procedures based on a direct discretization of the elliptic problem require the replacement of the condition on a finite artificial surface. Direct imposition of the condition at infinity along the finite boundary results in large errors. A sequence of boundary conditions is developed which provides increasingly accurate approximations to the problem in the infinite domain. Estimates of the error due to the finite boundary are obtained for several cases. Computations are presented which demonstrate the increased accuracy that can be obtained by the use of the higher order boundary conditions. The examples are based on a finite element formulation but finite difference methods can also be used

A fourth‐order compact scheme for the Helmholtz equation: Alpha‐interpolation of FEM and FDM stencils

We propose a fourth‐order compact scheme on structured meshes for the Helmholtz equation given by R(φ):=f(x)+Δφ+ξ2φ=0. The scheme consists of taking the alpha‐interpolation of the Galerkin finite element method and the classical central finite difference method. In 1D, this scheme is identical to the alpha‐interpolation method (J. Comput. Appl. Math. 1982; 8(1):15–19) and in 2D making the choice α=0.5 we recover the generalized fourth‐order compact Padé approximation (J. Comput. Phys. 1995; 119:252–270; Comput. Meth. Appl. Mech. Engrg 1998; 163:343–358) (therein using the parameter γ=2). We follow (SIAM Rev. 2000; 42(3):451–484; Comput. Meth. Appl. Mech. Engrg 1995; 128:325–359) for the analysis of this scheme and its performance on square meshes is compared with that of the quasi‐stabilized FEM (Comput. Meth. Appl. Mech. Engrg 1995; 128:325–359). In particular, we show that the relative phase error of the numerical solution and the local truncation error of this scheme for plane wave solutions diminish at the rate O((ξℓ)4), where ξ, ℓ represent the wavenumber and the mesh size, respectively. An expression for the parameter α is given that minimizes the maximum relative phase error in a sense that will be explained in Section 4.5. Convergence studies of the error in the L2 norm, the H1 semi‐norm and the l∞ Euclidean norm are done and the pollution effect is found to be small. Copyright © 2010 John Wiley & Sons, Ltd

A Numerical Minimization Scheme for the Complex Helmholtz Equation

We use the work of Milton, Seppecher, and Bouchitt\'{e} on variational principles for waves in lossy media to formulate a finite element method for solving the complex Helmholtz equation that is based entirely on minimization. In particular, this method results in a finite element matrix that is symmetric positive-definite and therefore simple iterative descent methods and preconditioning can be used to solve the resulting system of equations. We also derive an error bound for the method and illustrate the method with numerical experiments.Comment: 18 pages, 4 figure

A Petrov–Galerkin formulation for the alpha interpolation of FEM and FDM stencils: Applications to the Helmholtz equation

A new Petrov–Galerkin (PG) method involving two parameters, namely α1 and α2, is presented, which yields the following schemes on rectangular meshes: (i) a compact stencil obtained by the linear interpolation of the Galerkin FEM and the classical central finite difference method (FDM), should the parameters be equal, that is, α1 = α2 = α; and (ii) the nonstandard compact stencil presented in (Int. J. Numer. Meth. Engng 2011; 86:18–46) for the Helmholtz equation if the parameters are distinct, that is, α1 ≠ α2. The nonstandard compact stencil is obtained by taking the linear interpolation of the diffusive terms (specified by α1) and the mass terms (specified by α2) that appear in the stencils obtained by the standard Galerkin FEM and the classical central FDM, respectively. On square meshes, these two schemes were shown to provide solutions to the Helmholtz equation that have a dispersion accuracy of fourth and sixth order, respectively (Int. J. Numer. Meth. Engng 2011; 86:18–46). The objective of this paper is to study the performance of this PG method for the Helmholtz equation using nonuniform meshes and the treatment of natural boundary conditions

Nonstandard Finite Element Methods

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Stabilization in relation to wavenumber in HDG methods

Simulation of wave propagation through complex media relies on proper understanding of the properties of numerical methods when the wavenumber is real and complex. Numerical methods of the Hybrid Discontinuous Galerkin (HDG) type are considered for simulating waves that satisfy the Helmholtz and Maxwell equations. It is shown that these methods, when wrongly used, give rise to singular systems for complex wavenumbers. A sufficient condition on the HDG stabilization parameter for guaranteeing unique solvability of the numerical HDG system, both for Helmholtz and Maxwell systems, is obtained for complex wavenumbers. For real wavenumbers, results from a dispersion analysis are presented. An asymptotic expansion of the dispersion relation, as the number of mesh elements per wave increase, reveal that some choices of the stabilization parameter are better than others. To summarize the findings, there are values of the HDG stabilization parameter that will cause the HDG method to fail for complex wavenumbers. However, this failure is remedied if the real part of the stabilization parameter has the opposite sign of the imaginary part of the wavenumber. When the wavenumber is real, values of the stabilization parameter that asymptotically minimize the HDG wavenumber errors are found on the imaginary axis. Finally, a dispersion analysis of the mixed hybrid Raviart-Thomas method showed that its wavenumber errors are an order smaller than those of the HDG method