Volume 6, Issue 3
An Efficient Numerical Solution Method for Elliptic Problems in Divergence Form

Adv. Appl. Math. Mech., 6 (2014), pp. 327-344.

Published online: 2014-06

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• Abstract

In this paper the problem $-{\rm div}(a(x,y)\nabla u)=f$ 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 Abbas and Croisille (J. Sci. Comput., 49 (2011), pp. 239--267).

• Keywords

Hermitian scheme, box-scheme, Kronecker product, fast solver, iterative method, Poisson problem.

65N35, 65N08

• BibTex
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@Article{AAMM-6-327, author = {}, title = {An Efficient Numerical Solution Method for Elliptic Problems in Divergence Form}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2014}, volume = {6}, number = {3}, pages = {327--344}, abstract = {

In this paper the problem $-{\rm div}(a(x,y)\nabla u)=f$ 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 Abbas and Croisille (J. Sci. Comput., 49 (2011), pp. 239--267).

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.2012.m69}, url = {http://global-sci.org/intro/article_detail/aamm/22.html} }
TY - JOUR T1 - An Efficient Numerical Solution Method for Elliptic Problems in Divergence Form JO - Advances in Applied Mathematics and Mechanics VL - 3 SP - 327 EP - 344 PY - 2014 DA - 2014/06 SN - 6 DO - http://doi.org/10.4208/aamm.2012.m69 UR - https://global-sci.org/intro/article_detail/aamm/22.html KW - Hermitian scheme, box-scheme, Kronecker product, fast solver, iterative method, Poisson problem. AB -

In this paper the problem $-{\rm div}(a(x,y)\nabla u)=f$ 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 Abbas and Croisille (J. Sci. Comput., 49 (2011), pp. 239--267).

Ali Abbas. (1970). An Efficient Numerical Solution Method for Elliptic Problems in Divergence Form. Advances in Applied Mathematics and Mechanics. 6 (3). 327-344. doi:10.4208/aamm.2012.m69
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