An Efficient Numerical Solution Method for Elliptic Problems in Divergence Form (Advances in Applied Mathematics and Mechanics, 2014)
In this paper the problem $-{\rm div}(a(x,y) abla u)=f$ with Dirichlet boundary conditions on a square issolved iteratively with high accuracy for $u$ and $ abla u$ using a new scheme called"hermitian box-scheme". The design of the scheme is based on a "hermitian box", combining theapproximation of the gradient by the fourth order hermitian derivative, with a conservative discreteformulation on boxes of length 2$h$.The iterative technique is based on the repeated solution by a fast direct method of a discrete Poissonequation on a uniform rectangular mesh. The problem is suitably scaled before iteration. The numericalresults obtained show the efficiency of the numerical scheme. This work is the extension to stronglyelliptic problems of the hermitian box-scheme presented by Abbas and Croisille (J. Sci. Comput., 49 (2011), pp. 239--267).