Volume 5, Issue 1
Second Order Multigrid Methods for Elliptic Problems with Discontinuous Coefficients on an Arbitrary Interface, I: One Dimensional Problems

Numer. Math. Theor. Meth. Appl., 5 (2012), pp. 19-42.

Published online: 2012-05

[An open-access article; the PDF is free to any online user.]

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• Abstract

In this paper we present a one dimensional second order accurate method to solve Elliptic equations with discontinuous coefficients on an arbitrary interface. Second order accuracy for the first derivative is obtained as well. The method is based on the Ghost Fluid Method, making use of ghost points on which the value is defined by suitable interface conditions. The multi-domain formulation is adopted, where the problem is split in two sub-problems and interface conditions will be enforced to close the problem. Interface conditions are relaxed together with the internal equations (following the approach proposed in [10] in the case of smooth coefficients), leading to an iterative method on all the set of grid values (inside points and ghost points). A multigrid approach with a suitable definition of the restriction operator is provided. The restriction of the defect is performed separately for both sub-problems, providing a convergence factor close to the one measured in the case of smooth coefficient and independent on the magnitude of the jump in the coefficient. Numerical tests will confirm the second order accuracy. Although the method is proposed in one dimension, the extension in higher dimension is currently underway [12] and it will be carried out by combining the discretization of [10] with the multigrid approach of [11] for Elliptic problems with non-eliminated boundary conditions in arbitrary domain.

35J25, 65N06, 65N55

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@Article{NMTMA-5-19, author = {}, title = {Second Order Multigrid Methods for Elliptic Problems with Discontinuous Coefficients on an Arbitrary Interface, I: One Dimensional Problems}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2012}, volume = {5}, number = {1}, pages = {19--42}, abstract = {

In this paper we present a one dimensional second order accurate method to solve Elliptic equations with discontinuous coefficients on an arbitrary interface. Second order accuracy for the first derivative is obtained as well. The method is based on the Ghost Fluid Method, making use of ghost points on which the value is defined by suitable interface conditions. The multi-domain formulation is adopted, where the problem is split in two sub-problems and interface conditions will be enforced to close the problem. Interface conditions are relaxed together with the internal equations (following the approach proposed in [10] in the case of smooth coefficients), leading to an iterative method on all the set of grid values (inside points and ghost points). A multigrid approach with a suitable definition of the restriction operator is provided. The restriction of the defect is performed separately for both sub-problems, providing a convergence factor close to the one measured in the case of smooth coefficient and independent on the magnitude of the jump in the coefficient. Numerical tests will confirm the second order accuracy. Although the method is proposed in one dimension, the extension in higher dimension is currently underway [12] and it will be carried out by combining the discretization of [10] with the multigrid approach of [11] for Elliptic problems with non-eliminated boundary conditions in arbitrary domain.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2011.m12si02}, url = {http://global-sci.org/intro/article_detail/nmtma/5926.html} }
TY - JOUR T1 - Second Order Multigrid Methods for Elliptic Problems with Discontinuous Coefficients on an Arbitrary Interface, I: One Dimensional Problems JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 19 EP - 42 PY - 2012 DA - 2012/05 SN - 5 DO - http://doi.org/10.4208/nmtma.2011.m12si02 UR - https://global-sci.org/intro/article_detail/nmtma/5926.html KW - Elliptic equation, discontinuous coefficient, second order accuracy, multigrid, arbitrary interface, jump conditions. AB -

In this paper we present a one dimensional second order accurate method to solve Elliptic equations with discontinuous coefficients on an arbitrary interface. Second order accuracy for the first derivative is obtained as well. The method is based on the Ghost Fluid Method, making use of ghost points on which the value is defined by suitable interface conditions. The multi-domain formulation is adopted, where the problem is split in two sub-problems and interface conditions will be enforced to close the problem. Interface conditions are relaxed together with the internal equations (following the approach proposed in [10] in the case of smooth coefficients), leading to an iterative method on all the set of grid values (inside points and ghost points). A multigrid approach with a suitable definition of the restriction operator is provided. The restriction of the defect is performed separately for both sub-problems, providing a convergence factor close to the one measured in the case of smooth coefficient and independent on the magnitude of the jump in the coefficient. Numerical tests will confirm the second order accuracy. Although the method is proposed in one dimension, the extension in higher dimension is currently underway [12] and it will be carried out by combining the discretization of [10] with the multigrid approach of [11] for Elliptic problems with non-eliminated boundary conditions in arbitrary domain.

Armando Coco & Giovanni Russo. (2020). Second Order Multigrid Methods for Elliptic Problems with Discontinuous Coefficients on an Arbitrary Interface, I: One Dimensional Problems. Numerical Mathematics: Theory, Methods and Applications. 5 (1). 19-42. doi:10.4208/nmtma.2011.m12si02
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