MIXED FINITE ELEMENT METHODS FOR PROBLEMS .MIXED FINITE ELEMENT METHODS FOR PROBLEMS WITH ROBIN BOUNDARY

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  • Helsinki University of Technology Institute of Mathematics Research Reports

    Espoo 2009 A580

    MIXED FINITE ELEMENT METHODS FOR PROBLEMS

    WITH ROBIN BOUNDARY CONDITIONS

    Juho Konno Dominik Schotzau Rolf Stenberg

    AB TEKNILLINEN KORKEAKOULUTEKNISKA HGSKOLAN

    HELSINKI UNIVERSITY OF TECHNOLOGY

    TECHNISCHE UNIVERSITT HELSINKI

    UNIVERSITE DE TECHNOLOGIE DHELSINKI

  • Helsinki University of Technology Institute of Mathematics Research Reports

    Espoo 2009 A580

    MIXED FINITE ELEMENT METHODS FOR PROBLEMS

    WITH ROBIN BOUNDARY CONDITIONS

    Juho Konno Dominik Schotzau Rolf Stenberg

    Helsinki University of Technology

    Faculty of Information and Natural Sciences

    Department of Mathematics and Systems Analysis

  • Juho Konno, Dominik Schotzau, Rolf Stenberg: Mixed finite element meth-ods for problems with Robin boundary conditions; Helsinki University of TechnologyInstitute of Mathematics Research Reports A580 (2009).

    Abstract: We derive new a-priori and a-posteriori error estimates formixed finite element discretizations of second-order elliptic problems with gen-eral Robin boundary conditions, parameterized by a non-negative and piece-wise constant function. The estimates are robust over several orders of mag-nitude of the parameter, ranging from pure Dirichlet conditions to pure Neu-mann conditions. A series of numerical experiments is presented that verifyour theoretical results.

    AMS subject classifications: 65N30

    Keywords: mixed finite element methods, Robin boundary conditions, postpro-cessing, a posteriori analysis

    Correspondence

    Juho KonnoHelsinki University of TechnologyDepartment of Mathematics and Systems AnalysisP.O. Box 110002015 TKK, Finlandjkonno@math.tkk.fi

    Dominik SchotzauUniversity of British ColumbiaMathematics DepartmentVancouver, BCV6T 1Z2, Canadaschoetzau@math.ubc.ca

    Rolf StenbergHelsinki University of TechnologyDepartment of Mathematics and Systems AnalysisP.O. Box 110002015 TKK, Finlandrolf.stenberg@tkk.fi

    Received 2009-11-21ISBN 978-952-248-159-7 (print) ISSN 0784-3143 (print)ISBN 978-952-248-160-3 (PDF) ISSN 1797-5867 (PDF)

    Helsinki University of TechnologyFaculty of Information and Natural SciencesDepartment of Mathematics and Systems AnalysisP.O. Box 1100, FI-02015 TKK, Finland

    email: math@tkk.fi http://math.tkk.fi/

    http://math.tkk.fi/

  • MIXED FINITE ELEMENT METHODS FOR PROBLEMS WITH

    ROBIN BOUNDARY CONDITIONS

    JUHO KNN, DOMINIK SCHTZAU , AND ROLF STENBERG

    Abstract. We derive new a-priori and a-posteriori error estimates for mixed nite elementdiscretizations of second-order elliptic problems with general Robin boundary conditions, parameter-ized by a non-negative and piecewise constant function 0. The estimates are robust over severalorders of magnitude of , ranging from pure Dirichlet conditions to pure Neumann conditions. Aseries of numerical experiments is presented that verify our theoretical results.

    1. Introduction. We consider the dual mixed nite element method for secondorder elliptic equations subject to general Robin boundary conditions. These weparameterize by 0, with natural Dirichlet conditions for = 0 and essentialNeumann conditions in the limit . For the mixed method the Neumanncondition is an essential condition and could be explicitly enforced. However, weprefer to see the method implemented in the same way for all possible boundaryconditions and then the Neumann condition is obtained by penalization, i.e. choosing "large".

    Let us recall that the situation for a primal (displacement) nite element methodis the opposite, Neumann conditions are natural and Dirichlet essential, and the latterare penalized by choosing "small". For this case it is well known that the problembecomes ill-conditioned in two ways. The error estimates are not independent of and the stiness matrix becomes ill-conditioned. We here remark that in [8] Nitsche'smethod was extended to general Robin boundary conditions yielding a primal formu-lation avoiding ill-conditioning.

    The following question naturally arises now. Is the mixed method ill-conditionednear the Neumann limit? In this paper we will show that this is not the case. Wewill prove both a priori and a posteriori error estimates that are uniformly validindependent of the parameter . We also show that the stiness matrix is well-conditioned. It seems that this has not earlier been reported in the literature. Robinconditions are treated in [12], but the robustness with respect to the parameter wasnot studied.

    The outline of the paper is the following. In the next section we recall the mixednite element method. In Section 3 we derive a-priori error estimates and prove anoptimal bound for the error in the ux. In Section 4 we analyze the postprocessingmethod of [15, 14] which enhances the accuracy of the displacement variable. InSection 5, we introduce a residual-based a-posteriori error estimator and establish itsreliability and eciency. In Section 6 we consider the solution of the problem byhybridization and show that this leads to a well-conditioned linear system. A setof numerical examples are presented in Section 7 that verify the -robustness of ourestimates. Finally, we end the paper with some concluding remarks in Section 8.

    Throughout the paper, we use standard notation. We denote by C, C1, C2 etc.generic constants that are not necessarily identical at dierent places, but are always

    Institute of Mathematics, Helsinki University of Technology, P.O. Box 1100, FIN-02015 TKK,Finland (jkonno@math.tkk.fi).Mathematics Department, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

    (schoetzau@math.ubc.ca)Institute of Mathematics, Helsinki University of Technology, P.O. Box 1100, FIN-02015 TKK,

    Finland (rstenber@cc.hut.fi).

    1

  • independent of and the mesh size.

    2. Mixed nite element methods. In this section, we introduce two familiesof mixed nite element methods for the mixed form of Poisson's equation with Robinboundary conditions.

    2.1. Model problem. We consider the following model problem:

    u = 0 in , (2.1)div + f = 0 in , (2.2)

    subject to the general Robin boundary conditions

    n = u0 u+ g on . (2.3)Here, Rn, n = 2, 3, is a bounded polygonal or polyhedral Lipschitz domain, fa given load, and u0 and g are prescribed data on the boundary of . The vector ndenotes the unit outward normal vector on . The boundary conditions (2.3) areparameterized by the non-negative function 0. For simplicity, we assume to bepiecewise constant on the boundary (with respect to the partition of induced bya triangulation of ). In the limiting case = 0, we obtain the Dirichlet boundaryconditions

    u = u0 on . (2.4)

    On the other hand, if everywhere on , we recover the Neumann boundaryconditions

    n = g on . (2.5)To cast (2.1)(2.2) in weak form, we rst note that (, u) satises

    (, ) + (div , u) u, n = 0 H(div,), (2.6)(div , v) + (f, v) = 0 v L2(). (2.7)

    Then we solve for u in the expression (2.3) for the boundary conditions and insert theresult into (2.6). We nd that

    a(, ) + (div , u) = u0 + g, n H(div,), (2.8)(div , v) + (f, v) = 0 v L2(), (2.9)

    with a(, ) dened by

    a(, ) = (, ) + n, n.Here, we denote by (, ) the standard L2-inner product over , and by , the oneover the boundary . By introducing the bilinear form

    B(, u; , v) = a(, ) + (div , u) + (div , v),we thus obtain the following weak form of (2.1)(2.2): nd (, u) H(div,)L2()such that

    B(, u; , v) + (f, v) = u0 + g, n (2.10)for all ( , v) H(div,) L2().

    The well-posedness of (2.10) follows from standard arguments of mixed niteelement theory [4].

    2

  • 2.2. Mixed nite element discretization. In order to discretize the vari-ational problem (2.10), let Kh be a regular and shape-regular partition of intosimplices. As usual, the diameter of an element K is denoted by hK , and the globalmesh size h is dened as h = maxKKh hK . We denote by E0h the set of all interioredges (faces) of Kh, and by Eh the set of all boundary edges (faces). We write hE forthe diameter of an edge (face) E.

    Mixed nite element discretization of (2.10) is based on nite element spacesSh Vh H(div,) L2() of piecewise polynomial functions with respect to Kh.We will focus here on the Raviart-Thomas (RT) and Brezzi-Douglas-Marini (BDM)families of elements [11, 10, 3, 2, 4]. That is, for an approximation of order k 1, theux space Sh is taken as one of the following two spaces

    SRTh = { H(div,) ||K [Pk1(K)]n xPk1(K), K Kh },

    SBDMh = { H(div,) ||K [Pk(K)]n, K Kh },(2.11)

    where Pk(K) denotes the polynomials of total degree less or equal than k on K, andPk1(K) the homogeneous polynomials of degree k1. For both choices of Sh above,the displacements are approximated in the multiplier space

    Vh = {u L2() |u|K Pk1(K), K Kh }. (2.12)The spaces are chosen such that the following equilibrium property holds:

    div Sh Vh. (2.13)The mixed nite element method now consists of nding (h, uh) ShVh such that

    B(h, uh; , v) + (f, v) = u0 + g, n (2.14)for all ( , v) ShVh. We remark that, by the equilibrium condition (2.13), we haveimmediately the identity

    div h = Phf, (2.15)with Ph denoting the L

    2-projection onto Vh.

    3. A-priori error estimates. In this section, we derive a-priori error estimatesfor the method in (2.14).

    3.1. Stability. We begin by introducing the jump of a piecewise smooth scalarfunction u. To that end, let E = K K be an interior edge (face) shared by twoelements K and K . Then the jump of f over E is dened by

    [[f ]] = f |K f |K . (3.1)Next, we recall the following well-known trace estimate: for an edge (face) E of

    an element K, there holds

    hE20,E C20,K