Solving the 3D Laplace equation by meshless collocation via harmonic kernels

Hon, Y.C.; Schaback, Robert

doi:10.1007/s10444-011-9224-1

Cited by 11 publications

(5 citation statements)

References 8 publications

(12 reference statements)

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“…The direct meshless local Petrov-Galerkin (DMLPG) method is applied in [38] to the numerical solution of transient heat conduction problem. The main aim of [27] is to solve the Laplace equation ∆u = 0 on domains Ω ⊂ R 3 by meshless collocation on scattered points of the boundary ∂Ω. Authors of [21] provided a new way to compute and evaluate Gaussian radial basis function interpolants in a stable way with a special focus on small values of the shape parameter, i.e., for " flat" kernels.…”

Section: A Brief Review Of the Meshless Methodsmentioning

confidence: 99%

The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations on non-rectangular domains with error estimate

Dehghan

Abbaszadeh

Mohebbi

2015

Journal of Computational and Applied Mathematics

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Please cite this article as: M. Dehghan, M. Abbaszadeh, A. Mohebbi, The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin-Bona-Mahony-Burgers and regularized long-wave equations on non-rectangular domains with error estimate, Journal of Computational and Applied Mathematics (2015), http://dx. AbstractIn this paper a numerical technique is proposed for solving the nonlinear generalized Benjamin-Bona-Mahony-Burgers and regularized long-wave equations. Firstly, we obtain a time discrete scheme by approximating time derivative via a finite difference formula, then we use the interpolating element-free Galerkin approach to approximate the spatial derivatives. The element-free Galerkin method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the element-free Galerkin method test and trial functions are moving least squares approximation shape functions. Also, in the element-free Galekin method, we do not use any triangular, quadrangular or other type of meshes. It is a global method while finite elements method is a local one. The element free Galerkin method is not a truly meshless method and for integration employs a background mesh. We prove that the time discrete scheme is unconditionally stable and convergent in time variable using the energy method. We show that convergence order of the time discrete scheme is O(τ ). Since the shape functions of moving least squares approximation don't have Kronecker delta property, we can not implement the essential boundary condition, directly. Thus, the improved moving least squares shape functions that have the mentioned property are employed . An error estimate for the method proposed in the current paper is obtained. Also, the two-dimensional version of both equations on different complex geometries is solved. The aim of this paper is to show that the meshless method based on the weak form is also suitable for the treatment of the nonlinear partial differential equations and to obtain an error bound for the new method. Numerical examples confirm the efficiency of the proposed scheme. (M. Dehghan), a mohebbi@kashanu.ac.ir (A. Mohebbi), m.abbaszadeh@aut.ac.ir (M. Abbaszadeh).

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Section: A Brief Review Of the Meshless Methodsmentioning

confidence: 99%

The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations on non-rectangular domains with error estimate

Dehghan

Abbaszadeh

Mohebbi

2015

Journal of Computational and Applied Mathematics

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“…Due to its optimality properties, it is impossible to be outperformed error-wise for the given data, but is has serious stability and complexity drawbacks that are hard to overcome. A special case, connected to Trefftz methods and confined to potential problems, is in [32,15], but it deserves extensions using new kernels implementing singularity-free homogeneous solutions of other differential operators.…”

Section: Direct Optimal Recoverymentioning

confidence: 99%

“…This is a nasty approximation problem that should get much more attention by Approximation Theorists. The papers [32,15] use special kernel-based trial spaces where these approximation errors can be calculated, without any fictitious boundary. For the special case of equidistant points on concentric circles and conformal images of such configurations, results of Katsurada [19,20], handle the problem nicely by Fourier analysis.…”

Section: Trefftz Problemsmentioning

confidence: 99%

An Approximation Theorist’s view on solving operator equations—With special attention to Trefftz, MFS, MPS, and DRM methods

Schaback

2021

Computers & Mathematics with Applications

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“…Using the standard Hilbert space background [19,12], we also can formulate a P-greedy method for solving Dirichlet problems for harmonic functions in 2D or 3D. It will follow the Kolmogoroff N-width theory for such cases, but the latter seems to be open.…”

Section: Summary and Open Problemsmentioning

confidence: 99%

A Greedy Method for Solving Classes of PDE Problems

Schaback¹

2019

Preprint

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Motivated by the successful use of greedy algorithms for Reduced Basis Methods, a greedy method is proposed that selects N input data in an asymptotically optimal way to solve well-posed operator equations using these N data. The operator equations are defined as infinitely many equations given via a compact set of functionals in the dual of an underlying Hilbert space, and then the greedy algorithm, defined directly in the dual Hilbert space, selects N functionals step by step. When N functionals are selected, the operator equation is numerically solved by projection onto the span of the Riesz representers of the functionals. Orthonormalizing these yields useful Reduced Basis functions. By recent results on greedy methods in Hilbert spaces, the convergence rate is asymptotically given by Kolmogoroff N-widths and therefore optimal in that sense. However, these N-widths seem to be unknown in PDE applications. Numerical experiments show that for solving elliptic second-order Dirichlet problems, the greedy method of this paper behaves like the known P-greedy method for interpolation, applied to second derivatives. Since the latter technique is known to realize Kolmogoroff N-widths for interpolation, it is hypothesized that the Kolmogoroff N-widths for solving second-order PDEs behave like the Kolmogoroff N-widths for second derivatives, but this is an open theoretical problem.

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Solving the 3D Laplace equation by meshless collocation via harmonic kernels

Cited by 11 publications

References 8 publications

The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations on non-rectangular domains with error estimate

The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations on non-rectangular domains with error estimate

An Approximation Theorist’s view on solving operator equations—With special attention to Trefftz, MFS, MPS, and DRM methods

A Greedy Method for Solving Classes of PDE Problems

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