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Virtual Element Implementation for General Elliptic Equations

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Book cover Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 114))

Abstract

In the present paper we detail the implementation of the Virtual Element Method for two dimensional elliptic equations in primal and mixed form with variable coefficients.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Applications, University of Milano-Bicocca, via Cozzi 57, 20125, Milano, Italy

    Lourenco Beirão da Veiga & Alessandro Russo

  2. IMATI-CNR, via Ferrata 1, 27100, Pavia, Italy

    Lourenco Beirão da Veiga, Luisa Donatella Marini & Alessandro Russo

  3. Istituto di Matematica Applicata e Tecnologie Informatiche - Pavia, Consiglio Nazionale delle Ricerche, Pavia, Italy

    Franco Brezzi

  4. Department of Mathematics, University of Pavia, via Ferrata 1, 27100, Pavia, Italy

    Luisa Donatella Marini

Authors
  1. Lourenco Beirão da Veiga
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  2. Franco Brezzi
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  3. Luisa Donatella Marini
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  4. Alessandro Russo
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Corresponding author

Correspondence to Alessandro Russo .

Editor information

Editors and Affiliations

  1. Department of Mathematics & Statistics, University of Strathclyde, Glasgow, United Kingdom

    Gabriel R. Barrenechea

  2. Istituto di Matematica Applicata e Tecnologie Informatiche - Pavia, Consiglio Nazionale delle Ricerche, Pavia, Italy

    Franco Brezzi

  3. Department of Mathematics, University of Leicester, Leicester, United Kingdom

    Andrea Cangiani

  4. Department of Mathematics, National Technical University of Athens, Zografou, Greece

    Emmanuil H. Georgoulis

Appendix

Appendix

We list here the basis \(\boldsymbol{g}_{\alpha }^{\nabla,k}\) and \(\boldsymbol{g}_{\gamma }^{\oplus,k}\) obtained with MATLAB for k up to 5. We point out that in order to have the right scaling, the variable x and y must be replaced by \(\Big(\dfrac{x - x_{c}} {h_{E}} \Big)\) and \(\Big(\dfrac{x - y_{c}} {h_{E}} \Big)\) respectively.

\(\boldsymbol{g}_{\alpha }^{\nabla,k}\) \(\boldsymbol{g}_{\gamma }^{\oplus,k}\)

k=1     [         1,         0]    [             -y,       x]

        [         0,         1]

        [       2*x,         0]

        [         y,         x]

        [         0,       2*y]

k=2     [     3*x^2,         0]    [       -(x*y)/2,     x^2]

        [     2*x*y,       x^2]    [         -2*y^2,     x*y]

        [       y^2,     2*x*y]

        [         0,     3*y^2]

k=3     [     4*x^3,         0]    [     -(x^2*y)/3,     x^3]

        [   3*x^2*y,       x^3]    [         -x*y^2,   x^2*y]

        [   2*x*y^2,   2*x^2*y]    [         -3*y^3,   x*y^2]

        [       y^3,   3*x*y^2]

        [         0,     4*y^3]

k=4     [     5*x^4,         0]    [     -(x^3*y)/4,     x^4]

        [   4*x^3*y,       x^4]    [ -(2*x^2*y^2)/3,   x^3*y]

        [ 3*x^2*y^2,   2*x^3*y]    [   -(3*x*y^3)/2, x^2*y^2]

        [   2*x*y^3, 3*x^2*y^2]    [         -4*y^4,   x*y^3]

        [       y^4,   4*x*y^3]

        [         0,     5*y^4]

k=5     [     6*x^5,         0]    [     -(x^4*y)/5,     x^5]

        [   5*x^4*y,       x^5]    [   -(x^3*y^2)/2,   x^4*y]

        [ 4*x^3*y^2,   2*x^4*y]    [       -x^2*y^3, x^3*y^2]

        [ 3*x^2*y^3, 3*x^3*y^2]    [       -2*x*y^4, x^2*y^3]

        [   2*x*y^4, 4*x^2*y^3]    [         -5*y^5,   x*y^4]

        [       y^5,   5*x*y^4]

        [         0,     6*y^5]

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da Veiga, L.B., Brezzi, F., Marini, L.D., Russo, A. (2016). Virtual Element Implementation for General Elliptic Equations. In: Barrenechea, G., Brezzi, F., Cangiani, A., Georgoulis, E. (eds) Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations. Lecture Notes in Computational Science and Engineering, vol 114. Springer, Cham. https://doi.org/10.1007/978-3-319-41640-3_2

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