Weak Galerkin methods for second order elliptic interface problems
Weak Galerkin methods refer to general finite element methods for partial differential equations (PDEs) in which differential operators are approximated by their weak forms as distributions. Such weak forms give rise to desirable flexibilities in enforcing boundary and interface conditions. A weak G...
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| Published in | Journal of computational physics Vol. 250; pp. 106 - 125 |
|---|---|
| Main Authors | , , , , |
| Format | Journal Article |
| Language | English |
| Published |
United States
Elsevier Inc
01.10.2013
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0021-9991 1090-2716 |
| DOI | 10.1016/j.jcp.2013.04.042 |
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| Abstract | Weak Galerkin methods refer to general finite element methods for partial differential equations (PDEs) in which differential operators are approximated by their weak forms as distributions. Such weak forms give rise to desirable flexibilities in enforcing boundary and interface conditions. A weak Galerkin finite element method (WG-FEM) is developed in this paper for solving elliptic PDEs with discontinuous coefficients and interfaces. Theoretically, it is proved that high order numerical schemes can be designed by using the WG-FEM with polynomials of high order on each element. Extensive numerical experiments have been carried out to validate the WG-FEM for solving second order elliptic interface problems. High order of convergence is numerically confirmed in both L2 and L∞ norms for the piecewise linear WG-FEM. Special attention is paid to solve many interface problems, in which the solution possesses a certain singularity due to the nonsmoothness of the interface. A challenge in research is to design nearly second order numerical methods that work well for problems with low regularity in the solution. The best known numerical scheme in the literature is of order O(h) to O(h1.5) for the solution itself in L∞ norm. It is demonstrated that the WG-FEM of the lowest order, i.e., the piecewise constant WG-FEM, is capable of delivering numerical approximations that are of order O(h1.75) to O(h2) in the L∞ norm for C1 or Lipschitz continuous interfaces associated with a C1 or H2 continuous solution. |
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| AbstractList | Weak Galerkin methods refer to general finite element methods for partial differential equations (PDEs) in which differential operators are approximated by their weak forms as distributions. Such weak forms give rise to desirable flexibilities in enforcing boundary and interface conditions. A weak Galerkin finite element method (WG-FEM) is developed in this paper for solving elliptic PDEs with discontinuous coefficients and interfaces. Theoretically, it is proved that high order numerical schemes can be designed by using the WG-FEM with polynomials of high order on each element. Extensive numerical experiments have been carried out to validate the WG-FEM for solving second order elliptic interface problems. High order of convergence is numerically confirmed in both L2 and L∞ norms for the piecewise linear WG-FEM. Special attention is paid to solve many interface problems, in which the solution possesses a certain singularity due to the nonsmoothness of the interface. A challenge in research is to design nearly second order numerical methods that work well for problems with low regularity in the solution. The best known numerical scheme in the literature is of order O(h) to O(h1.5) for the solution itself in L∞ norm. It is demonstrated that the WG-FEM of the lowest order, i.e., the piecewise constant WG-FEM, is capable of delivering numerical approximations that are of order O(h1.75) to O(h2) in the L∞ norm for C1 or Lipschitz continuous interfaces associated with a C1 or H2 continuous solution. Weak Galerkin methods refer to general finite element methods for partial differential equations (PDEs) in which differential operators are approximated by their weak forms as distributions. Such weak forms give rise to desirable flexibilities in enforcing boundary and interface conditions. A weak Galerkin finite element method (WG-FEM) is developed in this paper for solving elliptic PDEs with discontinuous coefficients and interfaces. Theoretically, it is proved that high order numerical schemes can be designed by using the WG-FEM with polynomials of high order on each element. Extensive numerical experiments have been carried to validate the WG-FEM for solving second order elliptic interface problems. High order of convergence is numerically confirmed in both L2 and L∞ norms for the piecewise linear WG-FEM. Special attention is paid to solve many interface problems, in which the solution possesses a certain singularity due to the nonsmoothness of the interface. A challenge in research is to design nearly second order numerical methods that work well for problems with low regularity in the solution. The best known numerical scheme in the literature is of order [Formula: see text] to [Formula: see text] for the solution itself in L∞ norm. It is demonstrated that the WG-FEM of the lowest order, i.e., the piecewise constant WG-FEM, is capable of delivering numerical approximations that are of order [Formula: see text] to [Formula: see text] in the L∞ norm for C1 or Lipschitz continuous interfaces associated with a C1 or H2 continuous solution.Weak Galerkin methods refer to general finite element methods for partial differential equations (PDEs) in which differential operators are approximated by their weak forms as distributions. Such weak forms give rise to desirable flexibilities in enforcing boundary and interface conditions. A weak Galerkin finite element method (WG-FEM) is developed in this paper for solving elliptic PDEs with discontinuous coefficients and interfaces. Theoretically, it is proved that high order numerical schemes can be designed by using the WG-FEM with polynomials of high order on each element. Extensive numerical experiments have been carried to validate the WG-FEM for solving second order elliptic interface problems. High order of convergence is numerically confirmed in both L2 and L∞ norms for the piecewise linear WG-FEM. Special attention is paid to solve many interface problems, in which the solution possesses a certain singularity due to the nonsmoothness of the interface. A challenge in research is to design nearly second order numerical methods that work well for problems with low regularity in the solution. The best known numerical scheme in the literature is of order [Formula: see text] to [Formula: see text] for the solution itself in L∞ norm. It is demonstrated that the WG-FEM of the lowest order, i.e., the piecewise constant WG-FEM, is capable of delivering numerical approximations that are of order [Formula: see text] to [Formula: see text] in the L∞ norm for C1 or Lipschitz continuous interfaces associated with a C1 or H2 continuous solution. Weak Galerkin methods refer to general finite element methods for partial differential equations (PDEs) in which differential operators are approximated by their weak forms as distributions. Such weak forms give rise to desirable flexibilities in enforcing boundary and interface conditions. A weak Galerkin finite element method (WG-FEM) is developed in this paper for solving elliptic PDEs with discontinuous coefficients and interfaces. Theoretically, it is proved that high order numerical schemes can be designed by using the WG-FEM with polynomials of high order on each element. Extensive numerical experiments have been carried out to validate the WG-FEM for solving second order elliptic interface problems. High order of convergence is numerically confirmed in both L2L2 and LaLa norms for the piecewise linear WG-FEM. Special attention is paid to solve many interface problems, in which the solution possesses a certain singularity due to the nonsmoothness of the interface. A challenge in research is to design nearly second order numerical methods that work well for problems with low regularity in the solution. The best known numerical scheme in the literature is of order O(h)O(h) to O(h1.5)O(h1.5) for the solution itself in LaLa norm. It is demonstrated that the WG-FEM of the lowest order, i.e., the piecewise constant WG-FEM, is capable of delivering numerical approximations that are of order O(h1.75)O(h1.75) to O(h2)O(h2) in the LaLa norm for C1C1 or Lipschitz continuous interfaces associated with a C1C1 or H2H2 continuous solution. Weak Galerkin methods refer to general finite element methods for partial differential equations (PDEs) in which differential operators are approximated by their weak forms as distributions. Such weak forms give rise to desirable flexibilities in enforcing boundary and interface conditions. A weak Galerkin finite element method (WG-FEM) is developed in this paper for solving elliptic PDEs with discontinuous coefficients and interfaces. Theoretically, it is proved that high order numerical schemes can be designed by using the WG-FEM with polynomials of high order on each element. Extensive numerical experiments have been carried to validate the WG-FEM for solving second order elliptic interface problems. High order of convergence is numerically confirmed in both L2 and L∞ norms for the piecewise linear WG-FEM. Special attention is paid to solve many interface problems, in which the solution possesses a certain singularity due to the nonsmoothness of the interface. A challenge in research is to design nearly second order numerical methods that work well for problems with low regularity in the solution. The best known numerical scheme in the literature is of order O(h) to O(h1.5) for the solution itself in L∞ norm. It is demonstrated that the WG-FEM of the lowest order, i.e., the piecewise constant WG-FEM, is capable of delivering numerical approximations that are of order O(h1.75) to O(h2) in the L∞ norm for C1 or Lipschitz continuous interfaces associated with a C1 or H2 continuous solution. Weak Galerkin methods refer to general finite element methods for partial differential equations (PDEs) in which differential operators are approximated by their weak forms as distributions. Such weak forms give rise to desirable flexibilities in enforcing boundary and interface conditions. A weak Galerkin finite element method (WG-FEM) is developed in this paper for solving elliptic PDEs with discontinuous coefficients and interfaces. Theoretically, it is proved that high order numerical schemes can be designed by using the WG-FEM with polynomials of high order on each element. Extensive numerical experiments have been carried out to validate the WG-FEM for solving second order elliptic interface problems. High order of convergence is numerically confirmed in both L(2) and L( infinity ) norms for the piecewise linear WG-FEM. Special attention is paid to solve many interface problems, in which the solution possesses a certain singularity due to the nonsmoothness of the interface. A challenge in research is to design nearly second order numerical methods that work well for problems with low regularity in the solution. The best known numerical scheme in the literature is of order O(h) to O(h(1.5)) for the solution itself in L( infinity ) norm. It is demonstrated that the WG-FEM of the lowest order, i.e., the piecewise constant WG-FEM, is capable of delivering numerical approximations that are of order O(h(1.75)) to O(h(2)) in the L( infinity ) norm for C(1) or Lipschitz continuous interfaces associated with a C(1) or H(2) continuous solution. Weak Galerkin methods refer to general finite element methods for partial differential equations (PDEs) in which differential operators are approximated by their weak forms as distributions. Such weak forms give rise to desirable flexibilities in enforcing boundary and interface conditions. A weak Galerkin finite element method (WG-FEM) is developed in this paper for solving elliptic PDEs with discontinuous coefficients and interfaces. Theoretically, it is proved that high order numerical schemes can be designed by using the WG-FEM with polynomials of high order on each element. Extensive numerical experiments have been carried to validate the WG-FEM for solving second order elliptic interface problems. High order of convergence is numerically confirmed in both and norms for the piecewise linear WG-FEM. Special attention is paid to solve many interface problems, in which the solution possesses a certain singularity due to the nonsmoothness of the interface. A challenge in research is to design nearly second order numerical methods that work well for problems with low regularity in the solution. The best known numerical scheme in the literature is of order [Formula: see text] to [Formula: see text] for the solution itself in norm. It is demonstrated that the WG-FEM of the lowest order, i.e., the piecewise constant WG-FEM, is capable of delivering numerical approximations that are of order [Formula: see text] to [Formula: see text] in the norm for or Lipschitz continuous interfaces associated with a or continuous solution. |
| Author | Wang, Junping Ye, Xiu Wei, Guowei Zhao, Shan Mu, Lin |
| AuthorAffiliation | Department of Applied Science, University of Arkansas at Little Rock, Little Rock, AR 72204 ( lxmu@ualr.edu ) Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487 ( szhao@bama.ua.edu ) Department of Mathematics, Michigan State University, East Lansing, MI 48824 ( wei@math.msu.edu ) Division of Mathematical Sciences, National Science Foundation, Arlington, VA 22230 ( jwang@nsf.gov ) Department of Mathematics and Statistics, University of Arkansas at Little Rock, Little Rock, AR 72204 ( xxye@ualr.edu ) |
| AuthorAffiliation_xml | – name: Division of Mathematical Sciences, National Science Foundation, Arlington, VA 22230 ( jwang@nsf.gov ) – name: Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487 ( szhao@bama.ua.edu ) – name: Department of Mathematics and Statistics, University of Arkansas at Little Rock, Little Rock, AR 72204 ( xxye@ualr.edu ) – name: Department of Mathematics, Michigan State University, East Lansing, MI 48824 ( wei@math.msu.edu ) – name: Department of Applied Science, University of Arkansas at Little Rock, Little Rock, AR 72204 ( lxmu@ualr.edu ) |
| Author_xml | – sequence: 1 givenname: Lin surname: Mu fullname: Mu, Lin email: linmu@mail.math.msu.edu organization: Department of Mathematics, Michigan State University, East Lansing, MI 48824, United States – sequence: 2 givenname: Junping surname: Wang fullname: Wang, Junping email: jwang@nsf.gov organization: Division of Mathematical Sciences, National Science Foundation, Arlington, VA 22230, United States – sequence: 3 givenname: Guowei surname: Wei fullname: Wei, Guowei email: wei@math.msu.edu organization: Department of Mathematics, Michigan State University, East Lansing, MI 48824, United States – sequence: 4 givenname: Xiu surname: Ye fullname: Ye, Xiu email: xxye@ualr.edu organization: Department of Mathematics and Statistics, University of Arkansas at Little Rock,Little Rock, AR 72204, United States – sequence: 5 givenname: Shan surname: Zhao fullname: Zhao, Shan email: szhao@bama.ua.edu organization: Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487, United States |
| BackLink | https://www.ncbi.nlm.nih.gov/pubmed/24072935$$D View this record in MEDLINE/PubMed |
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| Keywords | Low solution regularity Nonsmooth interface Finite element methods Second order elliptic interface problems Weak Galerkin method nonsmooth interface weak Galerkin method low solution regularity second order elliptic interface problems |
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| Snippet | Weak Galerkin methods refer to general finite element methods for partial differential equations (PDEs) in which differential operators are approximated by... |
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| SubjectTerms | Approximation Boundaries Finite element method Finite element methods Galerkin methods Low solution regularity Mathematical analysis Mathematical models Nonsmooth interface Norms Partial differential equations Second order elliptic interface problems Weak Galerkin method |
| Title | Weak Galerkin methods for second order elliptic interface problems |
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