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On the construction of discretizations of elliptic partial differential equations. (English) Zbl 0907.65117

This paper deals with a class of finite element (FE) collocation methods for the approximate numerical solution of elliptic partial differential equations (PDEs). The main motivation for considering these methods is their potential usefulness in the numerical study of nonlinear phenomena in PDEs by continuation and bifurcation techniques. The author considers only algorithmic aspects and discusses relatively simple equations and domains, in order to focus on the basic ideas of the discretization method and the solution procedure. For linear elliptic PDEs the methods are equivalently defined as a type of generalized finite difference methods. The finite difference formulation leads in a very natural way to the nested dissection procedure for solving the discretized equations. Some numerical results are presented.

MSC:

65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
65N06 Finite difference methods for boundary value problems involving PDEs

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