19 research outputs found
Generalized convolution quadrature with variable time stepping
In this paper, we will present a generalized convolution quadrature for solving linear parabolic and hyperbolic evolution equations. The original convolution quadrature method by Lubich works very nicely for equidistant time steps while the generalization of the method and its analysis to nonuniform time stepping is by no means obvious. We will introduce the generalized convolution quadrature allowing for variable time steps and develop a theory for its error analysis. This method opens the door for further development towards adaptive time stepping for evolution equations. As the main application of our new theory, we will consider the wave equation in exterior domains which is formulated as a retarded boundary integral equatio
Adaptive BEM with optimal convergence rates for the Helmholtz equation
We analyze an adaptive boundary element method for the weakly-singular and
hypersingular integral equations for the 2D and 3D Helmholtz problem. The
proposed adaptive algorithm is steered by a residual error estimator and does
not rely on any a priori information that the underlying meshes are
sufficiently fine. We prove convergence of the error estimator with optimal
algebraic rates, independently of the (coarse) initial mesh. As a technical
contribution, we prove certain local inverse-type estimates for the boundary
integral operators associated with the Helmholtz equation
Boundary element methods for Helmholtz problems with weakly imposed boundary conditions
We consider boundary element methods where the Calder\'on projector is used
for the system matrix and boundary conditions are weakly imposed using a
particular variational boundary operator designed using techniques from
augmented Lagrangian methods. Regardless of the boundary conditions, both the
primal trace variable and the flux are approximated. We focus on the imposition
of Dirichlet and mixed Dirichlet--Neumann conditions on the Helmholtz equation,
and extend the analysis of the Laplace problem from the paper \emph{Boundary
element methods with weakly imposed boundary conditions} to this case. The
theory is illustrated by a series of numerical examples.Comment: 27 page
Sharp high-frequency estimates for the Helmholtz equation and applications to boundary integral equations
We consider three problems for the Helmholtz equation in interior and
exterior domains in R^d (d=2,3): the exterior Dirichlet-to-Neumann and
Neumann-to-Dirichlet problems for outgoing solutions, and the interior
impedance problem. We derive sharp estimates for solutions to these problems
that, in combination, give bounds on the inverses of the combined-field
boundary integral operators for exterior Helmholtz problems.Comment: Version 3: 42 pages; improved exposition in response to referee
comments and added several reference
Wavenumber-explicit analysis for the Helmholtz h-BEM: error estimates and iteration counts for the Dirichlet problem
We consider solving the exterior Dirichlet problem for the Helmholtz equation with the h-version of the boundary element method (BEM) using the standard second-kind combined-field integral equations. We prove a new, sharp bound on how the number of GMRES iterations must grow with the wavenumber k to have the error in the iterative solution bounded independently of k as k→∞ when the boundary of the obstacle is analytic and has strictly positive curvature. To our knowledge, this result is the first-ever sharp bound on how the number of GMRES iterations depends on the wavenumber for an integral equation used to solve a scattering problem. We also prove new bounds on how h must decrease with k to maintain k-independent quasi-optimality of the Galerkin solutions as k→∞ when the obstacle is nontrapping
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 -explicit stability (including -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 and the
approximation order are selected such that is sufficiently small and
, 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
Decompositions of high-frequency Helmholtz solutions via functional calculus, and application to the finite element method
Over the last ten years, results from [Melenk-Sauter, 2010], [Melenk-Sauter,
2011], [Esterhazy-Melenk, 2012], and [Melenk-Parsania-Sauter, 2013] decomposing
high-frequency Helmholtz solutions into "low"- and "high"-frequency components
have had a large impact in the numerical analysis of the Helmholtz equation.
These results have been proved for the constant-coefficient Helmholtz equation
in either the exterior of a Dirichlet obstacle or an interior domain with an
impedance boundary condition.
Using the Helffer-Sj\"ostrand functional calculus, this paper proves
analogous decompositions for scattering problems fitting into the black-box
scattering framework of Sj\"ostrand-Zworski, thus covering Helmholtz problems
with variable coefficients, impenetrable obstacles, and penetrable obstacles
all at once.
In particular, these results allow us to prove new frequency-explicit
convergence results for (i) the -finite-element method applied to the
variable coefficient Helmholtz equation in the exterior of a Dirichlet
obstacle, when the obstacle and coefficients are analytic, and (ii) the
-finite-element method applied to the Helmholtz penetrable-obstacle
transmission problem
High-frequency estimates on boundary integral operators for the Helmholtz exterior Neumann problem
We study a commonly-used second-kind boundary-integral equation for solving the Helmholtz exterior Neumann problem at high frequency, where, writing Γ for the boundary of the obstacle, the relevant integral operators map L2(Γ) to itself. We prove new frequency-explicit bounds on the norms of both the integral operator and its inverse. The bounds on the norm are valid for piecewise-smooth Γ and are sharp, and the bounds on the norm of the inverse are valid for smooth Γ and are observed to be sharp at least when Γ is curved. Together, these results give bounds on the condition number of the operator on L2(Γ); this is the first time L2(Γ) condition-number bounds have been proved for this operator for obstacles other than balls