On finite element methods for the Neumann problem

Abstract

Summary.The Neumann problem for a second order elliptic equation with self-adjoint operator is considered, the unique solution of which is determined from projection onto unity. Two variational formulations of this problem are studied, which have a unique solution in the whole space. Discretization is done via the finite element method based on the Ritz process, and it is proved that the discrete solutions converge to one of the solutions of the continuous problem. Comparison of the two methods is done.

A practical finite element approximation of a semi-definite Neumann problem on a curved domain

“…The approximation of the penalised problem (2.19) is advocated by Molchanov and Galba (1985) and their analysis shows that the optimal rate of convergence in the H l norm is achieved if e=O(h k-l) and any variational crimes present in the method are such that…”

“…However, as we shall see in Sect. 2 the constraints required by Molchanov and Galba (1985) on the finite element approximation of the penalised problem in order to obtain the optimal rate of convergence in the H 1 norm are not satisfied by standard practical approximations, for example, isoparametric elements. Another approach is to adjust the data so that the discrete version of the compatibility condition is satisfied.…”

“…A standard finite element (or finite difference) approximation of the system (1.2) may not have a solution as the discrete version of the compatibility condition (1.2b) may not be satisfied. One technique to overcome this problem is to study a penalised version of (1.2) by incorporating the term eu on the left hand side, as studied by Molchanov and Galba (1985). However, as we shall see in Sect.…”