Abstract
Nodal schemes have been originally developed in numerical reactor calculation, especially in the area of neutron diffusion problems. Broadly speaking, they constitute accurate and fast methods sharing many attractive features of the finite element as well as of the finite difference methods. In a previous work, some of the simplest nodal schemes were identified with nonstandard nonconforming finite element schemes exhibiting $O(h)$ convergence in $H^1 $ norm. We present here a more general family of nodal schemes with $O(h^k )$ convergence for any positive integer k under appropriate smoothness assumptions. This new family is first introduced within a straightforward nonconforming finite element framework. Under special numerical quadrature schemes, we are then led to nodal schemes which can be obtained directly through basic physical principles. Finally, dimensionally reduced versions are obtained by transverse integration and stand as strong candidates to practical implementations of the alternating direction type.
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