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Survey of meshless and generalized finite element methods: A unified approach. (English) Zbl 1048.65105
The authors “address meshless methods and the closely related generalized finite element methods for solving linear elliptic equations using variational principles.” The presentation includes both sketches of proofs, with references, and detailed proofs from the authors’ own work. The term “meshless methods” in the title refers to methods that either avoid the use of any mesh at all or use a mesh in a minimal manner, such as for numerical integration. The mesh is not used in the construction of the finite-dimensional spaces used to approximate the solution.
Meshless methods begin with a weak formulation of the differential problem, just as classical finite element methods do. Classical finite element methods generally approximate the solution of the differential problem using, for example, piecewise polynomials supported on a small number of elements of the underlying mesh. Linear polynomials give rise to so-called hill functions. Meshless methods, in contrast, use functions called “particle functions” supported on a small region not related to any underlying mesh. About two-thirds of the paper is devoted to the construction of particle functions and their approximating properties.
The table of contents of the paper includes ten sections: (1) Introduction; (2) The model problem; (3) Approximation by local functions in \({\mathcal R}^n\): the \(h\)-version analysis; (4) Construction and selection of particle shape functions; (5) Superconvergence of the gradient of the solution in \(L_2\); (6) The generalized finite element method; (7) Solutions of elliptic boundary value problems; (8) Implementational aspects of meshless methods; (9) Examples; (10) Future challenges.
Of these, Sections Three, Four and Five represent more than half the paper.
Section Three begins with historical development of particle shape functions including the conditions they are assumed to satisfy. Concepts such as \((t,k)\)-regularity, reproduction and quasi-reproduction are discussed. Approximation theorems are presented and proved, some classical and some recent.
Section Four describes the construction of reproducing kernel particle (RKP) shape functions in the context of results in Section Three. These shape functions can be either uniformly or non-uniformly distribute problems have recently been developed by various authors. These have almost exclusively dealt with thed and are defined as translations and scalings of a single function. A recent result of the authors estimates interpolation error using RKP shape functions.
Section Five is devoted to a superconvergence result similar to ones available for the finite element method, but proved using weighted Sobolev spaces. The authors state, “The main idea of the proof or our superconvergence result is to show that locally the approximation error is asymptotically the same as the error in the interpolation of a polynomial of degree \(k+1\) by particle shape functions.”
Sections Six, Seven and Eight discuss formulation and implementation of the so-called generalized finite element method, a particular example of a meshless method based on particle functions that form a partition of unity. Attention is given to choice of particle shape functions using solutions of special locally-posed problems called “handbook problems” that are particularly well-suited to the problem at hand. Attention is also given to enforcing Dirichlet boundary conditions and performing the integrals necessary for the matrix system of equations. Section Nine presents some examples that illustrate the value of meshless methods for complex geometries and for physical systems with too much internal structure for explicit modelling.

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
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