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Volume 15, Issue 4
A Kernel-Free Boundary Integral Method for Variable Coefficients Elliptic PDEs

Wenjun Ying & Wei-Cheng Wang

Commun. Comput. Phys., 15 (2014), pp. 1108-1140.

Published online: 2014-04

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  • Abstract

This work proposes a generalized boundary integral method for variable coefficients elliptic partial differential equations (PDEs), including both boundary value and interface problems. The method is kernel-free in the sense that there is no need to know analytical expressions for kernels of the boundary and volume integrals in the solution of boundary integral equations. Evaluation of a boundary or volume integral is replaced with interpolation of a Cartesian grid based solution, which satisfies an equivalent discrete interface problem, while the interface problem is solved by a fast solver in the Cartesian grid. The computational work involved with the generalized boundary integral method is essentially linearly proportional to the number of grid nodes in the domain. This paper gives implementation details for a second-order version of the kernel-free boundary integral method in two space dimensions and presents numerical experiments to demonstrate the efficiency and accuracy of the method for both boundary value and interface problems. The interface problems demonstrated include those with piecewise constant and large-ratio coefficients and the heterogeneous interface problem, where the elliptic PDEs on two sides of the interface are of different types.

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@Article{CiCP-15-1108, author = {}, title = {A Kernel-Free Boundary Integral Method for Variable Coefficients Elliptic PDEs}, journal = {Communications in Computational Physics}, year = {2014}, volume = {15}, number = {4}, pages = {1108--1140}, abstract = {

This work proposes a generalized boundary integral method for variable coefficients elliptic partial differential equations (PDEs), including both boundary value and interface problems. The method is kernel-free in the sense that there is no need to know analytical expressions for kernels of the boundary and volume integrals in the solution of boundary integral equations. Evaluation of a boundary or volume integral is replaced with interpolation of a Cartesian grid based solution, which satisfies an equivalent discrete interface problem, while the interface problem is solved by a fast solver in the Cartesian grid. The computational work involved with the generalized boundary integral method is essentially linearly proportional to the number of grid nodes in the domain. This paper gives implementation details for a second-order version of the kernel-free boundary integral method in two space dimensions and presents numerical experiments to demonstrate the efficiency and accuracy of the method for both boundary value and interface problems. The interface problems demonstrated include those with piecewise constant and large-ratio coefficients and the heterogeneous interface problem, where the elliptic PDEs on two sides of the interface are of different types.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.170313.071113s}, url = {http://global-sci.org/intro/article_detail/cicp/7130.html} }
TY - JOUR T1 - A Kernel-Free Boundary Integral Method for Variable Coefficients Elliptic PDEs JO - Communications in Computational Physics VL - 4 SP - 1108 EP - 1140 PY - 2014 DA - 2014/04 SN - 15 DO - http://doi.org/10.4208/cicp.170313.071113s UR - https://global-sci.org/intro/article_detail/cicp/7130.html KW - AB -

This work proposes a generalized boundary integral method for variable coefficients elliptic partial differential equations (PDEs), including both boundary value and interface problems. The method is kernel-free in the sense that there is no need to know analytical expressions for kernels of the boundary and volume integrals in the solution of boundary integral equations. Evaluation of a boundary or volume integral is replaced with interpolation of a Cartesian grid based solution, which satisfies an equivalent discrete interface problem, while the interface problem is solved by a fast solver in the Cartesian grid. The computational work involved with the generalized boundary integral method is essentially linearly proportional to the number of grid nodes in the domain. This paper gives implementation details for a second-order version of the kernel-free boundary integral method in two space dimensions and presents numerical experiments to demonstrate the efficiency and accuracy of the method for both boundary value and interface problems. The interface problems demonstrated include those with piecewise constant and large-ratio coefficients and the heterogeneous interface problem, where the elliptic PDEs on two sides of the interface are of different types.

Wenjun Ying & Wei-Cheng Wang. (2020). A Kernel-Free Boundary Integral Method for Variable Coefficients Elliptic PDEs. Communications in Computational Physics. 15 (4). 1108-1140. doi:10.4208/cicp.170313.071113s
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