Fast Nyström Methods for Parabolic Boundary Integral Equations

Tausch, Johannes

doi:10.1007/978-3-642-25670-7_6

Cited by 7 publications

(10 citation statements)

References 41 publications

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“…To compute interactions of these clusters efficiently, the kernel (2.2) is approximated by an expression that separates the (x, t) and (y, τ) variables, for instance, by a multivariate Chebyshev interpolation. This replaces the kernel by an approximation G ap with error [14] (4.1)…”

Section: Bilinear Formmentioning

confidence: 99%

“…Note that the matrices on the left-and right-hand sides are dense and must be evaluated by a fast method to overcome a complexity estimate that grows quadratically with the number of unknowns. This is achieved by an adaptation of the parabolic fast multipole method (pFMM) [13,14] to Galerkin discretized boundary integral operators of the heat equation as described in [7] and [6]. Downloaded 07/17/15 to 165.123.34.86.…”

Section: Efficient Solution Procedurementioning

confidence: 99%

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An Efficient Galerkin Boundary Element Method for the Transient Heat Equation

Messner

Schanz

Tausch

2015

SIAM J. Sci. Comput.

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We present boundary integral representations of several initial boundary value problems related to the heat equation. A Galerkin discretization with piecewise constant functions in time and piecewise linear functions in space leads to optimal a priori error estimates, provided that the meshwidths in space and time satisfy ht = O(h 2 x ). Each time step involves the solution of a linear system, whose spectral condition number is independent of the refinement under the same assumption on the mesh. We show that if the parabolic multipole method is used to apply parabolic boundary integral operators, the overall complexity of the scheme is log-linear while preserving the convergence of the Galerkin discretization method. The theoretical estimates are confirmed numerically at the end of the paper. [1,9], and Costabel [4]. For classically used second kind integral equations, e.g., a double-layer potential approach for the Dirichlet problem and a single-layer potential ansatz for the Neumann problem, the compactness of these integral operators for smooth domains guarantees the well-posedness and provides the backbone for the analysis of numerical methods. However, in the case of nonsmooth domains and first kind integral equations the situation is more complicated. Brown [3] gave some first results on Lipschitz domains before Arnold and Noon [1] and Costabel [4] proved almost contemporaneously the boundedness and coercivity of the thermal single-layer operator. Furthermore, the latter paper showed the coercivity of the hypersingular operator and the boundedness of all thermal boundary integral operators in the appropriate anisotropic Sobolev space setting. Introduction. Boundary integral equations related to the heat equation have been studied in Pogorzelski [11], Brown [3], Arnold and NoonWith these results the analysis of Galerkin methods in space and time follows the well-known pattern of the elliptic theory, i.e., the Lax-Milgram lemma guarantees uniqueness and solvability of the corresponding operator equations and their Galerkin variational formulation. Using conforming finite-dimensional subspaces of the natural energy spaces, uniqueness and solvability translates directly to the discrete system, where Cea's lemma provides quasi-optimality. Using the approximation property of piecewise polynomial finite-dimensional ansatz spaces, the regularity of the boundary integral operators, and assuming certain regularity of the discretization, one can derive explicit error estimates in the energy norm as well as weaker and stronger norms.

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Section: Bilinear Formmentioning

confidence: 99%

Section: Efficient Solution Procedurementioning

confidence: 99%

An Efficient Galerkin Boundary Element Method for the Transient Heat Equation

Messner

Schanz

Tausch

2015

SIAM J. Sci. Comput.

Self Cite

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“…As a consequence, although algorithms are available which solve the heat equation by layer potentials in essentially linear complexity relative to the number of unknowns in the tensor product space U Γ ×I L (cf. [14,15,19,20]), there is no gain in the use of boundary integral equations. To overcome this obstruction, as in [5], we shall consider a Galerkin discretization in the sparse tensor product of the ansatz spaces V Γ Ls and V I Lt .…”

Section: Sparse Tensor Product Discretizationmentioning

confidence: 99%

“…Such methods have been developed recently for the layer potentials of the heat equation when using the full tensor product space, see e.g. [19,20], but for sparse tensor product spaces this is still an open problem.…”

Section: Introductionmentioning

confidence: 99%

A fast sparse grid based space–time boundary element method for the nonstationary heat equation

Harbrecht

Tausch

2018

Numer. Math.

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This article presents a fast sparse grid based space-time boundary element method for the solution of the nonstationary heat equation. We make an indirect ansatz based on the thermal single layer potential which yields a first kind integral equation. This integral equation is discretized by Galerkin's method with respect to the sparse tensor product of the spatial and temporal ansatz spaces. By employing the H-matrix and Toeplitz structure of the resulting discretized operators, we arrive at an algorithm which computes the approximate solution in a complexity that essentially corresponds to that of the spatial discretization. Nevertheless, the convergence rate is nearly the same as in case of a traditional discretization in full tensor product spaces.

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“…However, the corresponding system matrices are dense which leads to high computational and memory complexities of a standard BEM. Therefore, several fast and data-sparse algorithms (such as Fourier series and FFT [7,8], the parabolic FMM [9,10], or a fast sparse grid method [11]) have been developed to provide efficient solvers with almost linear complexity. The parabolic fast multipole method (pFMM) has originally been described for Nyström discretizations [9,10] and has been extended to Galerkin discretizations [12,13] later on.…”

Section: Introductionmentioning

confidence: 99%

A parallel fast multipole method for a space-time boundary element method for the heat equation

Watschinger¹,

Merta²,

Of³

et al. 2021

Preprint

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We present a novel approach to the parallelization of the parabolic fast multipole method for a space-time boundary element method for the heat equation. We exploit the special temporal structure of the involved operators to provide an efficient distributed parallelization with respect to time and with a one-directional communication pattern. On top, we apply a task-based shared memory parallelization and SIMD vectorization. In the numerical tests we observe high efficiencies of our parallelization approach.

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Fast Nyström Methods for Parabolic Boundary Integral Equations

Cited by 7 publications

References 41 publications

An Efficient Galerkin Boundary Element Method for the Transient Heat Equation

An Efficient Galerkin Boundary Element Method for the Transient Heat Equation

A fast sparse grid based space–time boundary element method for the nonstationary heat equation

A parallel fast multipole method for a space-time boundary element method for the heat equation

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