Hierarchical matrix techniques for low- and high-frequency Helmholtz problems

In this paper, we discuss the application of hierarchical matrix techniques to the solution of Helmholtz problems with large wave number κ in 2D. We consider the Brakhage-Werner integral formulation of the problem discretized by the Galerkin boundary-element method. The dense n × n Galerkin matrix arising from this approach is represented by a sum of an -matrix and an 2-matrix, two different hierarchical matrix formats. A well-known multipole expansion is used to construct the 2-matrix. We present a new approach to dealing with the numerical instability problems of this expansion: the parts of the matrix that can cause problems are approximated in a stable way by an -matrix. Algebraic recompression methods are used to reduce the storage and the complexity of arithmetical operations of the -matrix. Further, an approximate LU decomposition of such a recompressed -matrix is an effective preconditioner. We prove that the construction of the matrices as well as the matrix-vector product can be performed in almost linear time in the number of unknowns. Numerical experiments for scattering problems in 2D are presented, where the linear systems are solved by a preconditioned iterative metho

A PDE approach to fractional diffusion: a space-fractional wave equation

We study solution techniques for an evolution equation involving second order derivative in time and the spectral fractional powers, of order $s \in (0,1)$, of symmetric, coercive, linear, elliptic, second-order operators in bounded domains $\Omega$. We realize fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on the semi-infinite cylinder $\mathcal{C} = \Omega \times (0,\infty)$. We thus rewrite our evolution problem as a quasi-stationary elliptic problem with a dynamic boundary condition and derive space, time, and space-time regularity estimates for its solution. The latter problem exhibits an exponential decay in the extended dimension and thus suggests a truncation that is suitable for numerical approximation. We propose and analyze two fully discrete schemes. The discretization in time is based on finite difference discretization techniques: trapezoidal and leapfrog schemes. The discretization in space relies on the tensorization of a first-degree FEM in $\Omega$ with a suitable $hp$-FEM in the extended variable. For both schemes we derive stability and error estimates

Runge-Kutta convolution coercivity and its use for time-dependent boundary integral equations

A coercivity property of temporal convolution operators is an essential tool in the analysis of time-dependent boundary integral equations and their space and time discretisations. It is known that this coercivity property is inherited by convolution quadrature time discretisation based on A-stable multistep methods, which are of order at most two. Here we study the ques- tion as to which Runge–Kutta-based convolution quadrature methods inherit the convolution coercivity property. It is shown that this holds without any restriction for the third-order Radau IIA method, and on permitting a shift in the Laplace domain variable, this holds for all algebraically stable Runge– Kutta methods and hence for methods of arbitrary order. As an illustration, the discrete convolution coercivity is used to analyse the stability and convergence properties of the time discretisation of a non-linear boundary integral equation that originates from a non-linear scattering problem for the linear wave equation. Numerical experiments illustrate the error behaviour of the Runge–Kutta convolution quadrature time discretisation

Convolution quadrature for the wave equation with a nonlinear impedance boundary condition

A rarely exploited advantage of time-domain boundary integral equations compared to their frequency counterparts is that they can be used to treat certain nonlinear problems. In this work we investigate the scattering of acoustic waves by a bounded obstacle with a nonlinear impedance boundary condition. We describe a boundary integral formulation of the problem and prove without any smoothness assumptions on the solution the convergence of a full discretization: Galerkin in space and convolution quadrature in time. If the solution is sufficiently regular, we prove that the discrete method converges at optimal rates. Numerical evidence in 3D supports the theory

A multipole method for Schwarz-Christoffel mapping of polygons with thousands of sides

A method is presented for the computation of Schwarz-Christoffel maps to polygons with tens of thousands of vertices. Previously published algorithms have CPU time estimates of the order O(N^3) for the computation of a conformal map of a polygon with N vertices. This has been reduced to O(N log N) by the use of the Fast Multipole Method and Davis's method for solving the parameter problem. The method is illustrated by a number of examples, the largest of which has N approx 2 × 10^5