The INTERNODES Method for Non-conforming Discretizations of PDEs

doi:10.1007/s42967-019-00020-1

Communications on Applied Mathematics and Computation ›› 2019, Vol. 1 ›› Issue (3): 361-401.doi: 10.1007/s42967-019-00020-1

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The INTERNODES Method for Non-conforming Discretizations of PDEs

Paola Gervasio¹, Alfio Quarteroni^2,3

1 DICATAM, Università degli Studi di Brescia, Brescia, Italy;
2 MOX, Department of Mathematics, Politecnico di Milano, Milan, Italy;
3 Institute of Mathematics, École Polytechnique Fédérale de Lausanne(EPFL), Lausanne, Switzerland

收稿日期:2018-09-07 修回日期:2019-01-31 出版日期:2019-09-20 发布日期:2019-09-09
通讯作者: Paola Gervasio, Alfio Quarteroni E-mail:paola.gervasio@unibs.it;alfio.quarteroni@polimi.it

The INTERNODES Method for Non-conforming Discretizations of PDEs

Paola Gervasio¹, Alfio Quarteroni^2,3

1 DICATAM, Università degli Studi di Brescia, Brescia, Italy;
2 MOX, Department of Mathematics, Politecnico di Milano, Milan, Italy;
3 Institute of Mathematics, École Polytechnique Fédérale de Lausanne(EPFL), Lausanne, Switzerland

Received:2018-09-07 Revised:2019-01-31 Online:2019-09-20 Published:2019-09-09

摘要/Abstract

摘要： INTERNODES is a general purpose method to deal with non-conforming discretizations of partial differential equations on 2D and 3D regions partitioned into two or several disjoint subdomains. It exploits two intergrid interpolation operators, one for transfering the Dirichlet trace across the interfaces, and the other for the Neumann trace. In this paper, in every subdomain the original problem is discretized by either the finite element method (FEM) or the spectral element method (SEM or hp-FEM), using a priori non-matching grids and piecewise polynomials of different degrees. Other discretization methods, however, can be used. INTERNODES can also be applied to heterogeneous or multiphysics problems, that is, problems that feature different differential operators inside adjacent subdomains. For instance, in this paper we apply the INTERNODES method to a Stokes- Darcy coupled problem that models the filtration of fluids in porous media. Our results highlight the flexibility of the method as well as its optimal rate of convergence with respect to the grid size and the polynomial degree.

关键词: Domain decomposition, Non-conforming approximation, Non-conforming grids, Interpolation, Finite element method, hp-finite element method, Spectral element method

Abstract: INTERNODES is a general purpose method to deal with non-conforming discretizations of partial differential equations on 2D and 3D regions partitioned into two or several disjoint subdomains. It exploits two intergrid interpolation operators, one for transfering the Dirichlet trace across the interfaces, and the other for the Neumann trace. In this paper, in every subdomain the original problem is discretized by either the finite element method (FEM) or the spectral element method (SEM or hp-FEM), using a priori non-matching grids and piecewise polynomials of different degrees. Other discretization methods, however, can be used. INTERNODES can also be applied to heterogeneous or multiphysics problems, that is, problems that feature different differential operators inside adjacent subdomains. For instance, in this paper we apply the INTERNODES method to a Stokes- Darcy coupled problem that models the filtration of fluids in porous media. Our results highlight the flexibility of the method as well as its optimal rate of convergence with respect to the grid size and the polynomial degree.

Key words: Domain decomposition, Non-conforming approximation, Non-conforming grids, Interpolation, Finite element method, hp-finite element method, Spectral element method

中图分类号:

Paola Gervasio, Alfio Quarteroni. The INTERNODES Method for Non-conforming Discretizations of PDEs[J]. Communications on Applied Mathematics and Computation, 2019, 1(3): 361-401.

参考文献

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The INTERNODES Method for Non-conforming Discretizations of PDEs

The INTERNODES Method for Non-conforming Discretizations of PDEs

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