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Optimally accurate higher-order finite element methods for polytopial approximations of domains with smooth boundaries
Authors: James Cheung, Mauro Perego, Pavel Bochev and Max Gunzburger
Journal: Math. Comp. 88 (2019), 2187-2219
MSC (2010): Primary 65N30
DOI: https://doi.org/10.1090/mcom/3415
Published electronically: February 21, 2019
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Abstract | References | Similar Articles | Additional Information
Abstract: Meshing of geometric domains having curved boundaries by affine simplices produces a polytopial approximation of those domains. The resulting error in the representation of the domain limits the accuracy of finite element methods based on such meshes. On the other hand, the simplicity of affine meshes makes them a desirable modeling tool in many applications. In this paper, we develop and analyze higher-order accurate finite element methods that remain stable and optimally accurate on polytopial approximations of domains with smooth boundaries. This is achieved by constraining a judiciously chosen extension of the finite element solution on the polytopial domain to weakly match the prescribed boundary condition on the true geometric boundary. We provide numerical examples that highlight key properties of the new method and that illustrate the optimal - and -norm convergence rates.
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P. G. Ciarlet and P.-A. Raviart, Interpolation theory over curved elements, with applications to finite element methods, Comput. Methods Appl. Mech. Engrg. 1 (1972), 217-249. MR 0375801, https://doi.org/10.1016/0045-7825(72)90006-0
Bernardo Cockburn, Weifeng Qiu, and Manuel Solano, A priori error analysis for HDG methods using extensions from subdomains to achieve boundary conformity, Math. Comp. 83 (2014), no. 286, 665-699. MR 3143688, https://doi.org/10.1090/S0025-5718-2013-02747-0
Bernardo Cockburn and Manuel Solano, Solving Dirichlet boundary-value problems on curved domains by extensions from subdomains, SIAM J. Sci. Comput. 34 (2012), no. 1, A497-A519. MR 2890275, https://doi.org/10.1137/100805200
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A. Main and G. Scovazzi, The shifted boundary method for embedded domain computations. Part II: Linear advection-diffusion and incompressible Navier-Stokes equations, Internat. J. Solids and Structures, to appear.
Francis J. Narcowich, Joseph D. Ward, and Holger Wendland, Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting, Math. Comp. 74 (2005), no. 250, 743-763. MR 2114646, https://doi.org/10.1090/S0025-5718-04-01708-9
Ruben Sevilla, Sonia Fernández-Méndez, and Antonio Huerta, NURBS-enhanced finite element method (NEFEM). A seamless bridge between CAD and FEM, Arch. Comput. Methods Eng. 18 (2011), no. 4, 441-484. MR 2851386, https://doi.org/10.1007/s11831-011-9066-5
Gilbert Strang and George J. Fix, An Analysis of the Finite Element Method, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1973. MR 0443377
Zhimin Zhang and Ahmed Naga, A new finite element gradient recovery method: superconvergence property, SIAM J. Sci. Comput. 26 (2005), no. 4, 1192-1213. MR 2143481, https://doi.org/10.1137/S1064827503402837
O. Zienkiewicz, R. Taylor, and R. Taylor, The Finite Element Method, McGraw-Hill, London, 1977.
Retrieve articles in Mathematics of Computation with MSC (2010): 65N30
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James Cheung
Affiliation: Interdisciplinary Center for Applied Mathematics, Virginia Tech, Blacksburg, Virginia 24061
Email: jamescheung@vt.edu
Mauro Perego
Affiliation: Center for Computing Research, Sandia National Laboratories, Albuquerque, New Mexico 87123
Email: mperego@sandia.gov
Pavel Bochev
Email: pbboche@sandia.gov
Max Gunzburger
Affiliation: Department of Scientific Computing, Florida State University, Tallahassee, Florida 32309
Email: mgunzburger@fsu.edu
Received by editor(s): February 8, 2018
Received by editor(s) in revised form: October 28, 2018
Additional Notes: Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-NA-0003525.
This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research. Additionally, the first and fourth authors were supported by US Department of Energy grant DE-SC0009324 and US Air Force Office of Scientific Research grant FA9550-15-1-0001.
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