arXiv subjects

Johannes Tausch

Publications and source records attributed to Johannes Tausch.

10 recordsLinked to original sources

arXiv2026

Space-Time Galerkin Boundary Element Method for the Wave Equation

The space-time Galerkin discretization of retarded layer potentials leads to a linear system where the coefficients are expressed in terms of integrals over the ansatz and test elements. They require carefully designed quadrature schemes because the kernel is singular in the origin and is discontinuous across the hyperbolicity cone. This paper introduces a new integration approach that leads to a scheme that converges exponentially with the number of quadrature points. The key here is to consider the integration domain as a convex polytope and to devise a decomposition into the convex hulls of simpler polytopes which are parameterized such that the singularity and discontinuity occurs in a single variable. The method is implemented for piecewise constant elements and tested on a scattering problem with known analytic solution.

math.NA↗

arXiv2025

Quadrature for Singular Integrals over convex Polytopes

A new algorithm for the efficient numerical approximation of weakly singular integrals over convex polytopes is introduced. Such integrals appear in the Galerkin discretizations of integral equations and nonlocal partial differential equations. The polytope is decomposed into a number of convex hulls of a singular and regular face. This expresses the singularity in a single variable which is effectively handled by Gauss-Jacobi quadrature. The decomposition algorithm is applicable to general finite polytopes. The Cartesian product of two simplices and two cubes will be discussed as special cases and numerical examples will be presented to illustrate the convergence of the resulting quadrature scheme.

math.NA↗

arXiv2025

A fast mesh-free boundary integral method for two-phase flow with soluble surfactant

We present an accurate and efficient boundary integral (BI) method for simulating the deformation of drops and bubbles in Stokes flow with soluble surfactant. Soluble surfactant advects and diffuses in bulk fluids while adsorbing and desorbing at interfaces. Since the fluid velocity is coupled to the surfactant concentration, the advection-diffusion equation governing the bulk surfactant concentration $C$ is nonlinear, precluding the Green's function formulation necessary for a BI method. However, in the physically representative large Péclet number limit, an analytical reduction of the surfactant dynamics permits a Green's function formulation for $C$ as an Abel-type time-convolution integral at each Lagrangian interface point. A challenge in developing a practical numerical method based on this formulation is the fast evaluation of the time convolution, since the kernel depends on the time history of quantities at the interface, which is only found during the time-stepping process. To address this, we develop a novel, causal version of the Fast Multipole Method that reduces the computational cost from $O(P^2)$ for direct evaluation of the time convolution to $O(P \log_2^2 P)$ per surface grid point, where $P$ is the number of time steps. In the bulk phase, the resulting method is mesh-free and provides an accurate solution to the fully coupled moving interface problem with soluble surfactant. The approach extends naturally to a broader class of advection-diffusion problems in the high Péclet number regime.

physics.flu-dyn↗

arXiv2024

A hybrid interpolation ACA accelerated method for parabolic boundary integral operators

We consider piecewise polynomial discontinuous Galerkin discretizations of boundary integral reformulations of the heat equation. The resulting linear systems are dense and block-lower triangular and hence can be solved by block forward elimination. For the fast evaluation of the history part, the matrix is subdivided into a family of sub-matrices according to the temporal separation. Separated blocks are approximated by Chebyshev interpolation of the heat kernel in time. For the spatial variable, we propose an adaptive cross approximation (ACA) framework to obtain a data-sparse approximation of the entire matrix. We analyse how the ACA tolerance must be adjusted to the temporal separation and present numerical results for a benchmark problem to confirm the theoretical estimates.

math.NA↗

arXiv2022

A Node Elimination Algorithm for Cubature of High-Dimensional Polytopes

Node elimination is a numerical approach to obtain cubature rules for the approximation of multivariate integrals. Beginning with a known cubature rule, nodes are selected for elimination, and a new, more efficient rule is constructed by iteratively solving the moment equations. This paper introduces a new criterion for selecting which nodes to eliminate that is based on a linearization of the moment equation. In addition, a penalized iterative solver is introduced, that ensures that weights are positive and nodes are inside the integration domain. A strategy for constructing an initial quadrature rule for various polytopes in several space dimensions is described. High efficiency rules are presented for two, three and four dimensional polytopes. The new rules are compared with rules that are obtained by combining tensor products of one dimensional quadrature rules and domain transformations, as well as with known analytically constructed cubature rules.

math.NA↗

arXiv2021

A fast method for evaluating Volume potentials in the Galerkin boundary element method

Three algorithm are proposed to evaluate volume potentials that arise in boundary element methods for elliptic PDEs. The approach is to apply a modified fast multipole method for a boundary concentrated volume mesh. If $h$ is the meshwidth of the boundary, then the volume is discretized using nearly $O(h^{-2})$ degrees of freedom, and the algorithm computes potentials in nearly $O(h^{-2})$ complexity. Here nearly means that logarithmic terms of $h$ may appear. Thus the complexity of volume potentials calculations is of the same asymptotic order as boundary potentials. For sources and potentials with sufficient regularity the parameters of the algorithm can be designed such that the error of the approximated potential converges at any specified rate $O(h^p)$. The accuracy and effectiveness of the proposed algorithms are demonstrated for potentials of the Poisson equation in three dimensions.

math.NA↗

arXiv2021

A Cartesian FMM-accelerated Galerkin boundary integral Poisson-Boltzmann solver

The Poisson-Boltzmann model is an effective and popular approach for modeling solvated biomolecules in continuum solvent with dissolved electrolytes. In this paper, we report our recent work in developing a Galerkin boundary integral method for solving the Poisson-Boltzmann (PB) equation. The solver has combined advantages in accuracy, efficiency, and memory usage as it applies a well-posed boundary integral formulation to circumvent many numerical difficulties associated with the PB equation and uses an O(N) Cartesian Fast Multipole Method (FMM) to accelerate the GMRES iteration. In addition, special numerical treatments such as adaptive FMM order, block diagonal preconditioners, Galerkin discretization, and Duffy's transformation are combined to improve the performance of the solver, which is validated on benchmark Kirkwood's sphere and a series of testing proteins.

physics.comp-ph↗

arXiv2020

Boundary integral operators for the heat equation in time-dependent domains

This article provides a functional analytical framework for boundary integral equations of the heat equation in time-dependent domains. More specifically, we consider a non-cylindrical domain in space-time that is the $C^2$-diffeomorphic image of a cylinder, i.e., the tensor product of a time interval and a fixed domain in space. On the non-cylindrical domain, we introduce Sobolev spaces, trace lemmata and provide the mapping properties of the layer operators by mimicking the proofs of [M. Costabel, Boundary integral operators for the heat equation, Integral Equations and Operator Theory, 13(4):498-552, 1990]. Here it is critical that the Neumann trace requires a correction term for the normal velocity of the moving boundary. Therefore, one has to analyze the situation carefully.

math.AP↗

arXiv2019

On the numerical solution of a time-dependent shape optimization problem for the heat equation

This article is concerned with the solution of a time-dependent shape identification problem. Specifically we consider the heat equation in a domain, which contains a time-dependent inclusion of zero temperature. The objective is to detect this inclusion from the given temperature and heat flux at the exterior boundary of the domain. To this end, for a given temperature at the exterior boundary, the mismatch of the Neumann data is minimized. This time-dependent shape optimization problem is then solved by a gradient-based optimization method. Numerical results are presented which validate the present approach.

math.OC↗

arXiv2005

A spectral method for integral formulations of medium-frequency scattering problems

A fast method for the computation of layer potentials that arise in acoustic scattering is introduced. The principal idea is to split the singular kernel into a smooth and a local part. The potential due to the smooth part is computed efficiently using non-equispaced FFTs, the potential due to the local part is expanded as a series in the mollification parameter. The complexity of the approach is shown to be $O(n + κ^3 \log κ)$, where $n$ is the number of degrees of freedom in the discretization and $κ$ is the wave number. The constant factor in this asymptotic estimate is small since no singular surface integrals must be computed. Therefore the method is particularly efficient for medium-sized scatterers (50-100 wavelengths) that may have complicated geometry

math.NA↗