Analysis of a continuous finite element method for H(curl,div)-elliptic interface problem

Huoyuan Duan, Ping Lin, Roger C. E. Tan

    Research output: Contribution to journalArticle

    12 Citations (Scopus)

    Abstract

    In this paper, we develop a continuous finite element method for the curlcurl-grad div vector second-order elliptic problem in a three-dimensional polyhedral domain occupied by discontinuous nonhomogeneous anisotropic materials. In spite of the fact that the curlcurl-grad div interface problem is closely related to the elliptic interface problem of vector Laplace operator type, the continuous finite element discretization of the standard variational problem of the former generally fails to give a correct solution even in the case of homogeneous media whenever the physical domain has reentrant corners and edges. To discretize the curlcurl-grad div interface problem by the continuous finite element method, we apply an element-local L2 projector to the curl operator and a pseudo-local L2 projector to the div operator, where the continuous Lagrange linear element enriched by suitable element and face bubbles may be employed. It is shown that the finite element problem retains the same coercivity property as the continuous problem. An error estimate O(hr) in an energy norm is obtained between the analytical solution and the continuous finite element solution, where the analytical solution is in ?Ll=1(Hr(Ol))3 for some r?(1/2,1] due to the domain boundary reentrant corners and edges (e.g., nonconvex polyhedron) and due to the interfaces between the different material domains in O=?Ll=1Ol .
    Original languageEnglish
    Pages (from-to)1-37
    Number of pages37
    JournalNumerische Mathematik
    Early online date14 Oct 2012
    DOIs
    Publication statusPublished - 2013

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    Interface Problems
    Curl
    Elliptic Problems
    Finite Element Method
    Finite element method
    Coercive force
    Interfaces (computer)
    Projector
    Analytical Solution
    Second-order Elliptic Problems
    Anisotropic Material
    Coercivity
    Finite Element Solution
    Laplace Operator
    Finite Element Discretization
    Operator
    Variational Problem
    Lagrange
    Polyhedron
    Bubble

    Cite this

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    title = "Analysis of a continuous finite element method for H(curl,div)-elliptic interface problem",
    abstract = "In this paper, we develop a continuous finite element method for the curlcurl-grad div vector second-order elliptic problem in a three-dimensional polyhedral domain occupied by discontinuous nonhomogeneous anisotropic materials. In spite of the fact that the curlcurl-grad div interface problem is closely related to the elliptic interface problem of vector Laplace operator type, the continuous finite element discretization of the standard variational problem of the former generally fails to give a correct solution even in the case of homogeneous media whenever the physical domain has reentrant corners and edges. To discretize the curlcurl-grad div interface problem by the continuous finite element method, we apply an element-local L2 projector to the curl operator and a pseudo-local L2 projector to the div operator, where the continuous Lagrange linear element enriched by suitable element and face bubbles may be employed. It is shown that the finite element problem retains the same coercivity property as the continuous problem. An error estimate O(hr) in an energy norm is obtained between the analytical solution and the continuous finite element solution, where the analytical solution is in ?Ll=1(Hr(Ol))3 for some r?(1/2,1] due to the domain boundary reentrant corners and edges (e.g., nonconvex polyhedron) and due to the interfaces between the different material domains in O=?Ll=1Ol .",
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    Analysis of a continuous finite element method for H(curl,div)-elliptic interface problem. / Duan, Huoyuan; Lin, Ping; Tan, Roger C. E.

    In: Numerische Mathematik, 2013, p. 1-37.

    Research output: Contribution to journalArticle

    TY - JOUR

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    AU - Lin, Ping

    AU - Tan, Roger C. E.

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    N2 - In this paper, we develop a continuous finite element method for the curlcurl-grad div vector second-order elliptic problem in a three-dimensional polyhedral domain occupied by discontinuous nonhomogeneous anisotropic materials. In spite of the fact that the curlcurl-grad div interface problem is closely related to the elliptic interface problem of vector Laplace operator type, the continuous finite element discretization of the standard variational problem of the former generally fails to give a correct solution even in the case of homogeneous media whenever the physical domain has reentrant corners and edges. To discretize the curlcurl-grad div interface problem by the continuous finite element method, we apply an element-local L2 projector to the curl operator and a pseudo-local L2 projector to the div operator, where the continuous Lagrange linear element enriched by suitable element and face bubbles may be employed. It is shown that the finite element problem retains the same coercivity property as the continuous problem. An error estimate O(hr) in an energy norm is obtained between the analytical solution and the continuous finite element solution, where the analytical solution is in ?Ll=1(Hr(Ol))3 for some r?(1/2,1] due to the domain boundary reentrant corners and edges (e.g., nonconvex polyhedron) and due to the interfaces between the different material domains in O=?Ll=1Ol .

    AB - In this paper, we develop a continuous finite element method for the curlcurl-grad div vector second-order elliptic problem in a three-dimensional polyhedral domain occupied by discontinuous nonhomogeneous anisotropic materials. In spite of the fact that the curlcurl-grad div interface problem is closely related to the elliptic interface problem of vector Laplace operator type, the continuous finite element discretization of the standard variational problem of the former generally fails to give a correct solution even in the case of homogeneous media whenever the physical domain has reentrant corners and edges. To discretize the curlcurl-grad div interface problem by the continuous finite element method, we apply an element-local L2 projector to the curl operator and a pseudo-local L2 projector to the div operator, where the continuous Lagrange linear element enriched by suitable element and face bubbles may be employed. It is shown that the finite element problem retains the same coercivity property as the continuous problem. An error estimate O(hr) in an energy norm is obtained between the analytical solution and the continuous finite element solution, where the analytical solution is in ?Ll=1(Hr(Ol))3 for some r?(1/2,1] due to the domain boundary reentrant corners and edges (e.g., nonconvex polyhedron) and due to the interfaces between the different material domains in O=?Ll=1Ol .

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