FEM for elliptic eigenvalue problems: how coarse can the coarsest mesh be chosen? An experimental study

In this paper, we consider the numerical discretization of elliptic eigenvalue problems by Finite Element Methods and its solution by a multigrid method. From the general theory of finite element and multigrid methods, it is well known that the asymptotic convergence rates become visible only if the mesh width h is sufficiently small, h≤h 0. We investigate the dependence of the maximal mesh width h 0 on various problem parameters such as the size of the eigenvalue and its isolation distance. In a recent paper (Sauter in Finite elements for elliptic eigenvalue problems in the preasymptotic regime. Technical Report. Math. Inst., Univ. Zürich, 2007), the dependence of h 0 on these and other parameters has been investigated theoretically. The main focus of this paper is to perform systematic experimental studies to validate the sharpness of the theoretical estimates and to get more insights in the convergence of the eigenfunctions and -values in the preasymptotic regim

Numerical solution of exterior Maxwell problems by Galerkin BEM and Runge-Kutta convolution quadrature

In this paper we consider time-dependent electromagnetic scattering problems from conducting objects. We discretize the time-domain electric field integral equation using Runge-Kutta convolution quadrature in time and a Galerkin method in space. We analyze the involved operators in the Laplace domain and obtain convergence results for the fully discrete scheme. Numerical experiments indicate the sharpness of the theoretical estimate

Algoritmos eficientes para convoluciones adaptados a las aplicaciones

Proponemos algoritmos para la aproximacion de determinadas ecuaciones integrales de Volterra relevantes en las aplicaciones, que consiguen minimizar el coste computational, comprimir la memoria y son faciles de implementar. La novedad con respecto a otros metodos existentes en la literatura es que estos nuevos algoritmos utilizan mas informacion acerca del problema, estando diseñados para tratar familias especificas de aplicaciones. En primer lugar, consideraremos la aproximacion de integrales fraccionarias y, gracias a esto, la resolucion de ecuaciones diferenciales fraccionarias en tiempo. En segundo lugar, consideraremos la resolucion de ecuaciones de Schrödinger con potencial concentrado en un conjunto discreto de puntos. Tipicamente el analisis de estos problemas se realiza reformulando las ecuaciones como sistemas de ecuaciones integrales de Volterra. Para las dos familias de aplicaciones consideradas, proponemos una implementacion especial de las cuadraturas de convolucion de Lubich, en la que conseguimos comprimir enormemente la memoria, manteniendo el coste computacional al nivel de los mejores algoritmos propuestos hasta la fecha. Ademas los nuevos algoritmos simplifican enormemente la implementacion con respecto a los metodos pre-existentes y abren la puerta a implementaciones con paso variable. Los algoritmos que proponemos se basan en la inversion global (mediante una unica quadratura) de la transformada de Laplace en el intervalo de interes (0,T). La idea de base es una representacion especial de los pesos de convolucion y cuadraturas especiales para calcularlos. Finalmente, se mostraran resultados numericos que ilustraran el funcionamiento de esta nueva generacion de algoritmos.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

On stability of discretizations of the Helmholtz equation (extended version)

We review the stability properties of several discretizations of the Helmholtz equation at large wavenumbers. For a model problem in a polygon, a complete $k$-explicit stability (including $k$-explicit stability of the continuous problem) and convergence theory for high order finite element methods is developed. In particular, quasi-optimality is shown for a fixed number of degrees of freedom per wavelength if the mesh size $h$ and the approximation order $p$ are selected such that $kh/p$ is sufficiently small and $p = O(\log k)$, and, additionally, appropriate mesh refinement is used near the vertices. We also review the stability properties of two classes of numerical schemes that use piecewise solutions of the homogeneous Helmholtz equation, namely, Least Squares methods and Discontinuous Galerkin (DG) methods. The latter includes the Ultra Weak Variational Formulation

A space–time Trefftz discontinuous Galerkin method for the acoustic wave equation in first-order formulation

We introduce a space–time Trefftz discontinuous Galerkin method for the first-order transient acoustic wave equations in arbitrary space dimensions, extending the one-dimensional scheme of Kretzschmar et al. (IMA J Numer Anal 36:1599–1635, 2016). Test and trial discrete functions are space–time piecewise polynomial solutions of the wave equations. We prove well-posedness and a priori error bounds in both skeleton-based and mesh-independent norms. The space–time formulation corresponds to an implicit time-stepping scheme, if posed on meshes partitioned in time slabs, or to an explicit scheme, if posed on “tent-pitched” meshes. We describe two Trefftz polynomial discrete spaces, introduce bases for them and prove optimal, high-order h-convergence bounds

Revisiting the crowding phenomenon in Schwarz-Christoffel mapping

We address the problem of conformally mapping the unit disk to polygons with elongations. The elongations cause the derivative of the conformal map to be exponentially large in some regions. This crowding phenomenon creates difficulties in standard numerical methods for the computation of the conformal map. We make use of the Schwarz-Christoffel representation of the mapping and show that a simple change to the existing algorithms introduced by Trefethen [SIAM J. Sci. Statist. Comput., 1 (1980), pp. 82-102] makes it feasible to accurately compute conformal maps to polygons even in the presence of extreme crowding. For an efficient algorithm it is essential that a good initial guess for the parameters of the Schwarz-Christoffel map be available. A uniformly close initial guess can be obtained from the cross-ratios of certain quadrilaterals, as introduced in the CRDT algorithm of Driscoll and Vavasis [SIAM J. Sci. Comput., 19 (1998), pp. 1783-1803]. We present numerical experiments and compare our algorithms with the CRDT which has been particularly designed to combat crowding

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 {kappa} in 2D. We consider the Brakhage–Werner integral formulation of the problem discretized by the Galerkin boundary-element method. The dense n x n Galerkin matrix arising from this approach is represented by a sum of an Formula -matrix and an Formula 2-matrix, two different hierarchical matrix formats. A well-known multipole expansion is used to construct the Formula 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 Formula -matrix. Algebraic recompression methods are used to reduce the storage and the complexity of arithmetical operations of the Formula -matrix. Further, an approximate LU decomposition of such a recompressed Formula -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 method

Rapid solution of the wave equation in unbounded domains

In this paper we propose and analyze a new, fast method for the numerical solution of time domain boundary integral formulations of the wave equation. We employ Lubich's convolution quadrature method for the time discretization and a Galerkin boundary element method for the spatial discretization. The coefficient matrix of the arising system of linear equations is a triangular block Toeplitz matrix. Possible choices for solving the linear system arising from the above discretization include the use of fast Fourier transform (FFT) techniques and the use of data-sparse approximations. By using FFT techniques, the computational complexity can be reduced substantially while the storage cost remains unchanged and is, typically, high. Using data-sparse approximations, the gain is reversed; i.e., the computational cost is (approximately) unchanged while the storage cost is substantially reduced. The method proposed in this paper combines the advantages of these two approaches. First, the discrete convolution (related to the block Toeplitz system) is transformed into the (discrete) Fourier image, thereby arriving at a decoupled system of discretized Helmholtz equations with complex wave numbers. A fast data-sparse (e.g., fast multipole or panel-clustering) method can then be applied to the transformed system. Additionally, significant savings can be achieved if the boundary data are smooth and time-limited. In this case the right-hand sides of many of the Helmholtz problems are almost zero, and hence can be disregarded. Finally, the proposed method is inherently parallel. We analyze the stability and convergence of these methods, thereby deriving the choice of parameters that preserves the convergence rates of the unperturbed convolution quadrature. We also present numerical results which illustrate the predicted convergence behavior