An unfitted discontinuous Galerkin scheme for conservation laws on evolving surfaces
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Abstract
Motivated by considering partial differential equations arising from conservation laws posed on evolving surfaces, a new numerical method for an advection problem is developed and simple numerical tests are performed. The method is based on an unfitted discontinuous Galerkin approach where the surface is not explicitly tracked by the mesh which means the method is extremely flexible with respect to geometry. Furthermore, the discontinuous Galerkin approach is well-suited to capture the advection driven by the evolution of the surface without the need for a space-time formulation, back-tracking trajectories or streamline diffusion. The method is illustrated by a one-dimensional example and numerical results ar presented that show good convergence properties for a simple test problem.
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How to Cite
Engwer, C., Ranner, T., & Westerheide, S.
(2016).
An unfitted discontinuous Galerkin scheme for conservation laws on evolving surfaces.
Proceedings Of The Conference Algoritmy, , 44-54.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/393/310
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References
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[8] Klaus Deckelnick, Charles M Elliott, and Vanessa Styles. Numerical diffusion-induced grain boundary motion. Interfaces and Free Boundaries, 3(4):393–414, 2001.
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[13] Carston Eilks and Charles M. Elliott. Numerical simulation of dealloying by surface dissolution via the evolving surface finite element method. Journal of Computational Physics, 227(23):9727–9741, 2008.
[14] Charles M. Elliott, Björn Stinner, Vanessa Styles, and Richard Welford. Numerical computation of advection and diffusion on evolving diffuse interfaces. IMA Journal of Numerical Analysis, 31(3):786–812, 2011.
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[16] Charles M Elliott and Chandrasekhar Venkataraman. Error analysis for an ale evolving surface finite element method. Numerical Methods for Partial Differential Equations, 31(2):459– 499, 2015.
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[18] Ashley J. James and John Lowengrub. A surfactant-conserving volume-of-fluid method for interfacial flows with insoluble surfactant. Journal of Computational Physics, 201(2):685–722, December 2004.
[19] Matthew P. Neilson, John A. Mackenzie, Steven D. Webb, and Robert H. Insall. Modeling Cell Movement and Chemotaxis Using Pseduopod-Based Feedback. SIAM Journal on Scientific Computing, 33:1035–1057, 2011.
[20] Maxim A. Olshanskii, Arnold Reusken, and Jörg Grande. A finite element method for elliptic equations on surfaces. SIAM Journal on Numerical Analysis, 47(5):3339–3358, 2009.
[21] Maxim A. Olshanskii, Arnold Reusken, and Xianmin Xu. An eulerian space-time finite element method for diffusion problems on evolving surfaces. SIAM Journal on Numerical Analysis, 52(3):1354–1377, 2014.
[22] Maxim A. Olshanskii, Arnold Reusken, and Xianmin Xu. A stabilized finite element method for advection–diffusion equations on surfaces. IMA Journal of Numerical Analysis, 34(2):732– 758, 2014.
[23] Knut Erik Teigen, Xiangrong Li, John Lowengrub, Fan Wang, and Axel Voigt. A diffuse-interface approach for modeling transport, diffusion and adsorption/desorption of material quantities on a deformable interface. Communications in mathematical sciences, 4(7):1009, 2009.
[2] P. Bastian, M. Blatt, A. Dedner, C. Engwer, R. Klöfkorn , R. Kornhuber, M. Ohlberger, and O. Sander. A generic grid interface for parallel and adaptive scientific computing. Part II: Implementation and tests in DUNE. Computing, 82(2–3):121–138, 7 2008.
[3] P. Bastian and C. Engwer. An unfitted finite element method using discontinuous galerkin. International Journal for Numerical Methods in Engineering, 79(12):1557–1576, 2009.
[4] Peter Bastian, Christian Engwer, Jorrit Fahlke, and Olaf Ippisch. An unfitted discontinuous galerkin method for pore-scale simulations of solute transport. Mathematics and Computers in Simulation, 81(10):2051 – 2061, 2011. MAMERN 2009: 3rd International Conference on Approximation Methods and Numerical Modeling in Environment and Natural Resources.
[5] Paolo Cermelli, Eliot Fried, and Morton E. Gurtin. Transport relations for surface integrals arising in the formulation of balance laws for evolving fluid interfaces. Journal of Fluid Mechanics, 544:339–351, 12 2005.
[6] Klaus Deckelnick, Gerhard Dziuk, Charles M. Elliott, and Claus-Justus Heine. An h-narrow band finite-element method for elliptic equations on implicit surfaces. IMA Journal of Numerical Analysis, 30(2):351–376, 2010.
[7] Klaus Deckelnick, Charles M. Elliott, and Thomas Ranner. Unfitted finite element methods using bulk meshes for surface partial differential equations. SIAM Journal on Numerical Analysis, 52(4):2137–2162, 2014.
[8] Klaus Deckelnick, Charles M Elliott, and Vanessa Styles. Numerical diffusion-induced grain boundary motion. Interfaces and Free Boundaries, 3(4):393–414, 2001.
[9] Gerhard Dziuk. Finite elements for the Beltrami operator on arbitrary surfaces. In Stefan Hildebrandt and Rolf Leis, editors, Partial Differential Equations and Calculus of Variations, volume 1357 of Lecture Notes in Mathematics, pages 142–155. 1988.
[10] Gerhard Dziuk and Charles M. Elliott. Finite elements on evolving surfaces. IMA Journal of Numerical Analysis, 27(2):262–292, 2007.
[11] Gerhard Dziuk and Charles M. Elliott. Finite element methods for surface PDEs. Acta Numerica, 22:289–396, 2013.
[12] Gerhard Dziuk, Dietmar Kröner, and Thomas Müller. Scalar conservation laws on moving hypersurfaces. Interfaces and Free Boundaries, 15(2):203–236, 2013.
[13] Carston Eilks and Charles M. Elliott. Numerical simulation of dealloying by surface dissolution via the evolving surface finite element method. Journal of Computational Physics, 227(23):9727–9741, 2008.
[14] Charles M. Elliott, Björn Stinner, Vanessa Styles, and Richard Welford. Numerical computation of advection and diffusion on evolving diffuse interfaces. IMA Journal of Numerical Analysis, 31(3):786–812, 2011.
[15] Charles M. Elliott and Vanessa Styles. An ALE ESFEM for solving PDEs on evolving surfaces. Milan Journal of Mathematics, 80(2):469–501, 2012.
[16] Charles M Elliott and Chandrasekhar Venkataraman. Error analysis for an ale evolving surface finite element method. Numerical Methods for Partial Differential Equations, 31(2):459– 499, 2015.
[17] C. Engwer and F. Heimann. Dune-udg: A cut-cell framework for unfitted discontinuous galerkin methods. In Proceedings of the DUNE User Meeting 2010, 2011.
[18] Ashley J. James and John Lowengrub. A surfactant-conserving volume-of-fluid method for interfacial flows with insoluble surfactant. Journal of Computational Physics, 201(2):685–722, December 2004.
[19] Matthew P. Neilson, John A. Mackenzie, Steven D. Webb, and Robert H. Insall. Modeling Cell Movement and Chemotaxis Using Pseduopod-Based Feedback. SIAM Journal on Scientific Computing, 33:1035–1057, 2011.
[20] Maxim A. Olshanskii, Arnold Reusken, and Jörg Grande. A finite element method for elliptic equations on surfaces. SIAM Journal on Numerical Analysis, 47(5):3339–3358, 2009.
[21] Maxim A. Olshanskii, Arnold Reusken, and Xianmin Xu. An eulerian space-time finite element method for diffusion problems on evolving surfaces. SIAM Journal on Numerical Analysis, 52(3):1354–1377, 2014.
[22] Maxim A. Olshanskii, Arnold Reusken, and Xianmin Xu. A stabilized finite element method for advection–diffusion equations on surfaces. IMA Journal of Numerical Analysis, 34(2):732– 758, 2014.
[23] Knut Erik Teigen, Xiangrong Li, John Lowengrub, Fan Wang, and Axel Voigt. A diffuse-interface approach for modeling transport, diffusion and adsorption/desorption of material quantities on a deformable interface. Communications in mathematical sciences, 4(7):1009, 2009.