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The partition of unity method. (English) Zbl 0949.65117
A new type of finite element method (FEM), the so-called partition of unity method (PUM), is introduced. The method is intended for the solution of problems where classical FEM approaches fail, e.g., where the polynomials used as form functions have poor approximation properties. If knowledge about the exact solution is available, the PUM can be adapted by constructing suitable local approximation spaces. A theorem on the approximation properties of the PUM is formulated. A main feature of the PUM is the construction of ansatz spaces of any desired regularity.
The authors discuss the principles of the PUM on the basis of a one-dimensional example for which the choice of local approximation spaces is considered. A PUM in two dimensions is briefly outlined. Numerical results for Laplace and Helmholtz equations on the unit square are presented to illustrate the advantages of the method.
The paper is concluded by an a posteriori error estimation.

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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