Why Does MLPG Work?

This is a short summary of recent mathematical results on error bounds and convergence of certain unsymmetric methods, including variations of Kansa’s collocation technique and Atluri’s MLPG method. The presentation is kept as simple as possible in order to address a larger community working on applications in Science and Engineering. Introduction and Summary The Meshless Local Petrov-Galerkin method (MLPG) of S.N. Atluri and T.L. Zhu [3] [4] of 1998 has been applied widely and very successfully in recent years, and it led already to various surveys and two books [1] [2]. However, its rigorous mathematical analysis lagged far behind its success in Science and Engineering. The same holds for the unsymmetric collocation technique introduced as early as 1986 by E. Kansa [7], which is confined to problems in strong form and uses collocation for trial spaces generated by multiquadrics, a special kind of radial kernel. It can be viewed as a special case of the MLPG, and it was called MLPG2 in [1] [2]. Its mathematical analysis was attempted by the author and others for several years, but was finally bound to fail because of a counterexample [5] given in 2001, showing that solvability of the final linear system cannot be guaranteed in general. Changes of the original method are necessary, avoiding solvability problems. This argument applies also to the more general situation of the MLPG, but the easier case of strong collocation was tackled first. A non-quantitative convergence result for a variation of Kansa’s unsymmetric collocation method was given in [9], while [10] contained a general convergence theory for a class of unsymmetric methods in strong form. The paper [6] by Hu, Li, and Cheng deals with the important special case of solving elliptic problems with analytic solutions by unsymmetric collocation based on analytic radial basis functions, leading to exponential convergence rates. Handling unsymmetric problems in weak form turned out to be more complicated, because there was no satisfactory theory of weak testing, so far. After investigating weak approximation problems [11] without differential equations, it was finally possible [12] to deal with a large class of unsymmetric methods solving partial differential equations in weak form, including a variation of the MLPG. The cited papers [10] [12] contain the mathematical core of a general framework built for analysis of computational methods solving general linear operator equations by unsymmetric methods in strong or weak form. However, the presentation and the results are necessarily in a rigorous and abstract mathematical style, and they require a solid background in mathematics, including regularity theory of PDEs and nonstandard results of approximation theory. To address a wider audience interested in computational methods in Science and Engineering, this paper summarizes these results in a somewhat more application-oriented 1Institut für Numerische und Angewandte Mathematik, Universität Göttingen, Lotzestraße 16–18, 37083 Göttingen, Germany language, and taking a Poisson problem −∆u = fΩ in Ω⊂ IRd ∂u ∂n = fN in ΓN ⊂ ∂Ω u = fD in ΓD ⊆ ∂Ω (1) as a running example for explanation. The final result is that unsymmetric computational methods can be rigorously proven to converge at certain rates, if 1. the underlying problem can be written as a solvable well-posed linear operator equation which need not be elliptic, 2. the chosen scale of trial spaces can approximate the solution well, 3. differential equations and boundary conditions are tested via separated local weak or strong forms, leading to sufficiently many well-formulated linear test equations satisfying a stability condition, 4. the final overdetermined non-square linear system of test equations for trial functions is solved approximately by minimization of the discrete residuals. Note that Atluri’s MLPG method and Kansa’s collocation technique are special cases, and they can be proven to converge, if they are set up properly along the above lines. Furthermore, the framework allows very general trial spaces and test strategies. This is in line with the many variations of the MLPG method induced by different test and trial strategies (see pages 140-143 of [1]), but the paper [12] does not cover all variations, since it only shows how kernel-based trial and test strategies fit into the framework. This leaves plenty of leeway for future research. However, readers should be aware that items 3 and 4 above contain two major differences to the standard setting of the MLPG variants. Problems like (1) are viewed here as systems whose equations are always tested separately. This is standard for strong testing, but for weak testing it means that we do not use a single weak form combining the three equations. We rather stay with separate local weak forms to be tested separately. Item 3 leads to non-square linear systems consisting of linear weak or strong test equations whose number must be expected to be much larger than the number of unknowns on the trial side, i.e. the number of columns in the system matrix. This calls for solution methods like least-squares which keep the residuals of the equations small. The framework of [12] proves that good approximate solutions to these systems exist, and reasonable numerical methods will not overlook them. Convergence rates are mainly dependent on item 2. They cannot be positively influenced by testing, since they are a matter of the trial side. Testing cares for safety, while the trial side determines the attainable accuracy. If trial functions are chosen via translates of smooth kernels, convergence rates increase with the smoothness of the kernel and the solution. The rest of the paper will explain the above items one by one, adding more details.