We present a solver for the 2D high-frequency Helmholtz equation in heterogeneous acoustic media, with online parallel complexity that scales optimally as @math , where @math is the number of volume unknowns, and @math is the number of processors, as long as @math grows at most like a small fractional power of @math . The solver decomposes the domain into layers, and uses transmission conditions in boundary integral form to explicitly define "polarized traces", i.e., up- and down-going waves sampled at interfaces. Local direct solvers are used in each layer to precompute traces of local Green's functions in an embarrassingly parallel way (the offline part), and incomplete Green's formulas are used to propagate interface data in a sweeping fashion, as a preconditioner inside a GMRES loop (the online part). Adaptive low-rank partitioning of the integral kernels is used to speed up their application to interface data. The method uses second-order finite differences. The complexity scalings are empirical but motivated by an analysis of ranks of off-diagonal blocks of oscillatory integrals. They continue to hold in the context of standard geophysical community models such as BP and Marmousi 2, where convergence occurs in 5 to 10 GMRES iterations.
Their work spawned a series of new formulations within the framework of Discontinous Galerkin methods, such as the Trefftz formulation of @cite_25 @cite_4 , and the discontinuous enrichment method of @cite_75 ;
@cite_75 @cite_4 @cite_25
2100903519 1982359839 2127600768 2022811246
We present a compact discontinuous Galerkin (CDG) method for an elliptic model problem. The problem is first cast as a system of first order equations by introducing the gradient of the primal unknown, or flux, as an additional variable. A standard discontinuous Galerkin (DG) method is then applied to the resulting system of equations. The numerical interelement fluxes are such that the equations for the additional variable can be eliminated at the element level, thus resulting in a global system that involves only the original unknown variable. The proposed method is closely related to the local discontinuous Galerkin (LDG) method [B. Cockburn and C.-W. Shu, SIAM J. Numer. Anal., 35 (1998), pp. 2440-2463], but, unlike the LDG method, the sparsity pattern of the CDG method involves only nearest neighbors. Also, unlike the LDG method, the CDG method works without stabilization for an arbitrary orientation of the element interfaces. The computation of the numerical interface fluxes for the CDG method is slightly more involved than for the LDG method, but this additional complication is clearly offset by increased compactness and flexibility. Compared to the BR2 [F. Bassi and S. Rebay, J. Comput. Phys., 131 (1997), pp. 267-279] and IP [J. Douglas, Jr., and T. Dupont, in Computing Methods in Applied Sciences (Second Internat. Sympos., Versailles, 1975), Lecture Notes in Phys. 58, Springer, Berlin, 1976, pp. 207-216] methods, which are known to be compact, the present method produces fewer nonzero elements in the matrix and is computationally more efficient. We propose and analyze a discontinuous Galerkin approximation for the Stokes problem. The finite element triangulation employed is not required to be conforming and we use discontinuous pressures and velocities. No additional unknown fields need to be introduced, but only suitable bilinear forms defined on the interfaces between the elements, involving the jumps of the velocity and the average of the pressure. We consider hp approximations using ℚk′–ℚk velocity-pressure pairs with k′ = k + 2, k + 1, k. Our methods show better stability properties than the corresponding conforming ones. We prove that our first two choices of velocity spaces ensure uniform divergence stability with respect to the mesh size h. Numerical results show that they are uniformly stable with respect to the local polynomial degree k, a property that has no analog in the conforming case. An explicit bound in k which is not sharp is also proven. Numerical results show that if equal order approximation is chosen for the velocity and pressure, no spurious pressure modes are present but the method is not uniformly stable either with respect to h or k. We derive a priori error estimates generalizing the abstract theory of mixed methods. Optimal error estimates in h are proven. As for discontinuous Galerkin methods for scalar diffusive problems, half of the power of k is lost for p and hp pproximations independently of the divergence stability. We are concerned with a finite element approximation for time-harmonic wave propagation governed by the Helmholtz equation. The usually oscillatory behavior of solutions, along with numerical dispersion, render standard finite element methods grossly inefficient already in medium-frequency regimes. As an alternative, methods that incorporate information about the solution in the form of plane waves have been proposed. We focus on a class of Trefftz-type discontinuous Galerkin methods that employs trial and test spaces spanned by local plane waves. In this paper we give a priori convergence estimates for the h -version of these plane wave discontinuous Galerkin methods in two dimensions. To that end, we develop new inverse and approximation estimates for plane waves and use these in the context of duality techniques. Asymptotic optimality of the method in a mesh dependent norm can be established. However, the estimates require a minimal resolution of the mesh beyond what it takes to resolve the wavelength. We give numerical evidence that this requirement cannot be dispensed with. It reflects the presence of numerical dispersion. This paper reviews some known and proposes some new preconditioning methods for a number of discontinuous Galerkin (or DG) finite element approximations for elliptic problems of second order. Nested hierarchy of meshes is generally assumed. Our approach utilizes a general two-level scheme, where the finite element space for the DG method is decomposed into a subspace (viewed as an auxiliary or ‘coarse’ space), plus a correction which can be handled by a standard smoothing procedure. We consider three different auxiliary subspaces, namely, piecewise linear C0-conforming functions, piecewise linear functions that are continuous at the centroids of the edges faces (Crouzeix-Raviart finite elements) and piecewise constant functions over the finite elements. To support the theoretical results, we also present numerical experiments for 3-D model problem showing uniform convergence of the constructed methods. Copyright © 2006 John Wiley & Sons, Ltd.