Direct Meshless Local Petrov–Galerkin (DMLPG) method: A generalized MLS approximation

Mirzaei, Davoud; Schaback, Robert

doi:10.1016/j.apnum.2013.01.002

Cited by 91 publications

(57 citation statements)

References 14 publications

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“…On the side, we investigated the stabilization effect of shifted scaled polynomial bases. In a forthcoming paper ( [16]), the GMLS will be applied to enhance the computational efficiency of the meshless local Petrov-Galerkin (MLPG) method of Atluri and his colleagues ( [5,4,3]) significantly. † , ROBERT SCHABACK ‡, * , MEHDI DEHGHAN § …”

Section: Discussionmentioning

confidence: 99%

“…Note that this requires evaluation of λ on polynomials only, not on any shape functions. This can be used to accelerate certain meshless methods for solving PDEs, as will be demonstrated in a follow-up paper ( [16]) focusing on applications.…”

Section: Classical and Diffuse Derivativesmentioning

confidence: 99%

“…We prove optimal convergence rates for the diffuse derivatives and give numerical examples. A forthcoming paper by [16] will apply our results to make the Meshless Local Petrov-Galerkin method (MLPG) by [5,3,4] considerably more effective. This is the main motivation behind our approach.…”

Section: Introductionmentioning

confidence: 99%

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On generalized moving least squares and diffuse derivatives

Mirzaei¹,

Schaback

Dehghan³

2011

IMA Journal of Numerical Analysis

187

116

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Abstract. The Moving Least Squares (MLS) method provides an approximationû of a function u based solely on values u(x j ) of u on scattered "meshless" nodes x j . Derivatives of u are usually approximated by derivatives ofû. In contrast to this, we directly estimate derivatives of u from the data, without any detour via derivatives ofû. This is a generalized Moving Least Squares technique, and we prove that it produces diffuse derivatives as introduced by Nyroles et. al. in 1992. Consequently, these turn out to be efficient direct estimates of the true derivatives, without anything "diffuse" about them, and we prove optimal rates of convergence towards the true derivatives. Numerical examples confirm this, and we finally show how the use of shifted and scaled polynomials as basis functions in the generalized and standard MLS approximation stabilizes the algorithm.

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Section: Discussionmentioning

confidence: 99%

Section: Classical and Diffuse Derivativesmentioning

confidence: 99%

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On generalized moving least squares and diffuse derivatives

Mirzaei¹,

Schaback

Dehghan³

2011

IMA Journal of Numerical Analysis

187

116

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“…As a numerical example for a meshless method in weak form, we take a case from [37] concerning convergence of the MLPG5 method as described and analyzed in Sect. 13.6.…”

Section: Numerical Examplesmentioning

confidence: 99%

All well-posed problems have uniformly stable and convergent discretizations

Schaback

2015

Numer. Math.

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This paper considers a large class of linear operator equations, including linear boundary value problems for partial differential equations, and treats them as linear recovery problems for functions from their data. Well-posedness of the problem means that this recovery is continuous. Discretization recovers restricted trial functions from restricted test data, and it is well-posed or stable, if this restricted recovery is continuous. After defining a general framework for these notions, this paper proves that all well-posed linear problems have stable and refinable computational discretizations with a stability that is determined by the well-posedness of the problem and independent of the computational discretization, provided that sufficiently many test data are used. The solutions of discretized problems converge when enlarging the trial spaces, and the convergence rate is determined by how well the data of the function solving the analytic problem can be approximated by the data of the trial functions. This allows new and very simple proofs of convergence rates for generalized finite elements, symmetric and unsymmetric Kansa-type collocation, and other meshfree methods like Meshless Local Petrov-Galerkin techniques. It is also shown that for a fixed trial space, weak formulations have a slightly better convergence rate than strong formulations, but at the expense of numerical integration. Since convergence rates are reduced to those coming from Approximation Theory, and since trial spaces are arbitrary, this also covers various spectral and pseudospectral methods. All of this is illustrated by examples.Mathematics Subject Classification 65M12 · 65M70 · 65N12 · 65N35 · 65M15 · 65M22 · 65J10 · 65J20 · 35D30 · 35D35 · 35B65 · 41A25 · 41A63 B Robert Schaback

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“…[7,18]. Bypassing Moving Least Squares trial functions, direct methods in the context of Meshless Local Petrov Galerkin techniques are in [21,20], connected to diffuse derivatives [23]. For a mixture of kernel-based and MLS techniques, see [17].…”

Section: Discretizationmentioning

confidence: 99%