The perfectly matched layer (PML) absorbing boundary condition has proven to be very efficient from a numerical point of view for the elastic wave equation to absorb both body waves with nongrazing incidence and surface waves. However, at grazing incidence the classical discrete PML method suffers from large spurious reflections that make it less efficient for instance in the case of very thin mesh slices, in the case of sources located close to the edge of the mesh, and/or in the case of receivers located at very large offset. We demonstrate how to improve the PML at grazing incidence for the differential seismic wave equation based on an unsplit convolution technique. The improved PML has a cost that is similar in terms of memory storage to that of the classical PML. We illustrate the efficiency of this improved convolutional PML based on numerical benchmarks using a finite-difference method on a thin mesh slice for an isotropic material and show that results are significantly improved compared with the classical PML technique. We also show that, as the classical PML, the convolutional technique is intrinsically unstable in the case of some anisotropic materials.

An unsplit convolutional perfectly matched layer improved at grazing incidence for the seismic wave equation

State-of-the-art finite-element methods for time-harmonic acoustics governed by the Helmholtz equation are reviewed. Four major current challenges in the field are specifically addressed: the effective treatment of acoustic scattering in unbounded domains, including local and nonlocal absorbing boundary conditions, infinite elements, and absorbing layers; numerical dispersion errors that arise in the approximation of short unresolved waves, polluting resolved scales, and requiring a large computational effort; efficient algebraic equation solving methods for the resulting complex-symmetric (non-Hermitian) matrix systems including sparse iterative and domain decomposition methods; and a posteriori error estimates for the Helmholtz operator required for adaptive methods. Mesh resolution to control phase error and bound dispersion or pollution errors measured in global norms for large wave numbers in finite-element methods are described. Stabilized, multiscale, and other wave-based discretization methods developed to reduce this error are reviewed. A review of finite-element methods for acoustic inverse problems and shape optimization is also given.

A review of finite-element methods for time-harmonic acoustics

In contrast to the positive definite Helmholtz equation, the deceivingly similar looking indefinite Helmholtz equation is difficult to solve using classical iterative methods. Simply using a Krylov method is much less effective, especially when the wave number in the Helmholtz operator becomes large, and also algebraic preconditioners such as incomplete LU factorizations do not remedy the situation. Even more powerful preconditioners such as classical domain decomposition and multigrid methods fail to lead to a convergent method, and often behave differently from their usual behavior for positive definite problems. For example increasing the overlap in a classical Schwarz method degrades its performance, as does increasing the number of smoothing steps in multigrid. The purpose of this review paper is to explain why classical iterative methods fail to be effective for Helmholtz problems, and to show different avenues that have been taken to address this difficulty.

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Why it is Difficult to Solve Helmholtz Problems with Classical Iterative Methods

Higher-order finite element methods.

A fast algorithm is presented for solving electromagnetic scattering from a rectangular open cavity embedded in an infinite ground plane The medium inside the cavity is assumed to be (vertically) layered By introducing a transparent (artificial) boundary condition, the problem in the open cavity is reduced to a bounded domain problem A simple finite difference method is then applied to solve the model Helmholtz equation The fast algorithm is designed for solving the resulting discrete system in terms of the discrete Fourier transform in the horizontal direction, a Gaussian elimination in the vertical direction, and a preconditioning conjugate gradient method with a complex diagonal preconditioner for the indefinite interface system The existence and uniqueness of the finite difference solution are established for arbitrary wave numbers Our numerical experiments for large numbers of mesh points, up to 16 million unknowns, and for large wave numbers, eg, between 100 and 200 wavelengths, show that the algorithm is extremely efficient The cost for calculating the radar cross section, which is of significant interest in practice, is O(M2) for an $M \times M$ mesh The proposed algorithm may be extended easily to solve discrete systems from other discretization methods of the model problem

A Fast Algorithm for the Electromagnetic Scattering from a Large Cavity

http://users.jyu.fi/~tene/papers/reportB02-07.pdf

An algebraic multigrid based shifted-Laplacian preconditioner for the Helmholtz equation

Fictitious Domain Methods for the Numerical Solution of Two-Dimensional Scattering Problems

The application of a fictitious domain (domain embedding) method to the three-dimensional Helmholtz equation with absorbing boundary conditions is considered. The finite element discretization is performed by using locally fitted meshes, and an algebraic fictitious domain method with a separable preconditioner is used in the iterative solution of the resultant linear systems. Such a method is based on embedding the original domain into a larger one with a simple geometry. With this approach, it is possible to realize the GMRES iterations in a low-dimensional subspace and use the partial solution method to solve the linear systems with the preconditioner. An efficient parallel implementation of the iterative algorithm is introduced. Results of numerical experiments demonstrate good scalability properties on distributed-memory parallel computers and the ability to solve high frequency acoustic scattering problems.

Erkki Heikkola

Papers

An algebraic multigrid based shifted-Laplacian preconditioner for the Helmholtz equation

Fictitious Domain Methods for the Numerical Solution of Two-Dimensional Scattering Problems

A Parallel Fictitious Domain Method for the Three-Dimensional Helmholtz Equation.

A Parallel Fictitious Domain Method for the Three-Dimensional Helmholtz Equation

Controllability method for the Helmholtz equation with higher-order discretizations