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A constructive method for deriving finite elements of nodal type. (English) Zbl 0677.65101
So-called “nodal” methods were introduced in the context of nuclear engineering for solution of the neutron diffusion and transport equations. The methods are designed to provide solution information over relatively coarse meshes. In this paper the authors put nodal methods into the framework of finite element methods, formulate a calculus to apply the patch test, place several well-known nodal methods into their framework, and introduce a three dimensional nodal method as a natural application of their approach.
The authors characterize nodal methods as non-conforming finite element methods whose degrees of freedom are area- and edge-moments of the solution (in two dimensions) or volume- and face-moments (in three dimensions). This choice of unknowns is in contrast with the usual choice of solution values at specified points in each element. Only quadrilateral or hexahedral elements are considered. The shape functions are normalized Legendre polynomials.
In this context, the authors present a calculus for application of the patch test to determine that accuracy is not lost due to a deficiency in the space of approximating polynomials. They also present examples of nodal methods for which their approach applies. These methods include: Langenbuch-Maurer-Werner $$\Sigma$$ nodal schemes, Hennart nodal schemes, Raviart-Thomas mixed finite element schemes, Brezzi-Douglas-Marini mixed finite element schemes, as well as a three-dimensional generalization of the latter schemes.
Reviewer: Myron Sussman

##### MSC:
 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 74S05 Finite element methods applied to problems in solid mechanics 35Q99 Partial differential equations of mathematical physics and other areas of application 82C70 Transport processes in time-dependent statistical mechanics
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