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Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz problem. (English) Zbl 0955.65081

Summary: A new technique to solve elliptic linear partial differential equations (PDEs), called ultra weak variational formulation in this paper, was introduced by B. Després [C. R. Acad. Sci., Paris, Sér. I 318, No. 10, 939-944 (1994; Zbl 0806.35026)]. This paper is devoted to an evaluation of the potentialities of this technique. It is applied to a model wave problem, the two-dimensional Helmholtz problem. The new method is presented in three parts following the same style of presentation as the classical one of the finite element method, even though they are definitely conceptually different methods.
The first part is committed to the variational formulation and to the continuous problem. The second part defines the discretization process using a Galerkin procedure. The third part actually studies the efficiency of the technique from the order of convergence point of view. This is achieved using theoretical proofs and a series of numerical experiments. In particular, it is proven and shown the order of convergence is lower bounded by a linear function of the number of degrees of freedom. An application to scattering problems is presented in a fourth part.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
78A45 Diffraction, scattering
78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs

Citations:

Zbl 0806.35026
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