Abbas, Ali An efficient numerical solution method for elliptic problems in divergence form. (English) Zbl 1308.65181 Adv. Appl. Math. Mech. 6, No. 3, 327-344 (2014). Summary: In this paper the problem \(-\mathrm{div}(a(x,y)\nabla u)=f\) with Dirichlet boundary conditions on a square is solved iteratively with high accuracy for \(u\) and \(\nabla u\) using a new scheme called “Hermitian box-scheme”. The design of the scheme is based on a “Hermitian box”, combining the approximation of the gradient by the fourth order hermitian derivative, with a conservative discrete formulation on boxes of length 2\(h\). The iterative technique is based on the repeated solution by a fast direct method of a discrete Poisson equation on a uniform rectangular mesh. The problem is suitably scaled before iteration. The numerical results obtained show the efficiency of the numerical scheme. This work is the extension to strongly elliptic problems of the Hermitian box-scheme presented by A. Abbas and J.-P. Croisille [J. Sci. Comput. 49, No. 3, 239–267 (2011; Zbl 1348.65148)]. MSC: 65N06 Finite difference methods for boundary value problems involving PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 65F10 Iterative numerical methods for linear systems Keywords:Hermitian scheme; box-scheme; Kronecker product; fast solver; iterative method; Poisson problem; numerical result Citations:Zbl 1348.65148 PDFBibTeX XMLCite \textit{A. Abbas}, Adv. Appl. Math. Mech. 6, No. 3, 327--344 (2014; Zbl 1308.65181) Full Text: DOI