Optimally accurate higher-order finite element methods for polytopial approximations of domains with smooth boundaries
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- by James Cheung, Mauro Perego, Pavel Bochev and Max Gunzburger HTML | PDF
- Math. Comp. 88 (2019), 2187-2219 Request permission
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 $H^1$- and $L^2$-norm convergence rates.References
- Robert A. Adams and John J. F. Fournier, Sobolev spaces, 2nd ed., Pure and Applied Mathematics (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003. MR 2424078
- John W. Barrett and Charles M. Elliott, Finite element approximation of the Dirichlet problem using the boundary penalty method, Numer. Math. 49 (1986), no. 4, 343–366. MR 853660, DOI 10.1007/BF01389536
- L. Beiräo Da Veiga, A. Russo, and G. Vacca, The virtual element method with curved edges, arXiv preprint (2018).
- Pavel B. Bochev and Max D. Gunzburger, Least-squares finite element methods, Applied Mathematical Sciences, vol. 166, Springer, New York, 2009. MR 2490235, DOI 10.1007/b13382
- Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 3rd ed., Texts in Applied Mathematics, vol. 15, Springer, New York, 2008. MR 2373954, DOI 10.1007/978-0-387-75934-0
- Erik Burman, Peter Hansbo, and Mats G. Larson, A cut finite element method with boundary value correction, Math. Comp. 87 (2018), no. 310, 633–657. MR 3739212, DOI 10.1090/mcom/3240
- J. Cheung, M. Perego, and P. Bochev, New developments in using Schwarz methods for model coupling, in Center for Computing Research Summer Proceedings 2015, A. M. Bradley and M. L. Parks, eds., Technical Report SAND2016-0830R, Sandia National Laboratories, 2015, pp. 14–26.
- J. Cheung, M. Perego, P. Bochev, and M. Gunzburger, An optimally convergent coupling approach for interface problems approximated with higher-order finite elements, arXiv preprint (2017).
- Philippe G. Ciarlet, The finite element method for elliptic problems, Classics in Applied Mathematics, vol. 40, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)]. MR 1930132, DOI 10.1137/1.9780898719208
- Philippe G. Ciarlet, Linear and nonlinear functional analysis with applications, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2013. MR 3136903
- 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 375801, DOI 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, DOI 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, DOI 10.1137/100805200
- Bernardo Cockburn and Manuel Solano, Solving convection-diffusion problems on curved domains by extensions from subdomains, J. Sci. Comput. 59 (2014), no. 2, 512–543. MR 3188451, DOI 10.1007/s10915-013-9776-y
- J. Austin Cottrell, Thomas J. R. Hughes, and Yuri Bazilevs, Isogeometric analysis, John Wiley & Sons, Ltd., Chichester, 2009. Toward integration of CAD and FEA. MR 3618875, DOI 10.1002/9780470749081
- Monique Dauge, Elliptic boundary value problems on corner domains, Lecture Notes in Mathematics, vol. 1341, Springer-Verlag, Berlin, 1988. Smoothness and asymptotics of solutions. MR 961439, DOI 10.1007/BFb0086682
- I. Ergatoudis, B. Irons, and O. Zienkiewicz, Curved, isoparametric, quadrilateral elements for finite element analysis, Internat. J. Solids and Structures 4 (1968), 31–42.
- Alexandre Ern and Jean-Luc Guermond, Theory and practice of finite elements, Applied Mathematical Sciences, vol. 159, Springer-Verlag, New York, 2004. MR 2050138, DOI 10.1007/978-1-4757-4355-5
- I. Fried, Accuracy and condition of curved (isoparametric) finite elements, J. Sound and Vibration 31 (1973), 345–355.
- Pierre Grisvard, Elliptic problems in nonsmooth domains, Classics in Applied Mathematics, vol. 69, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. Reprint of the 1985 original [ MR0775683]; With a foreword by Susanne C. Brenner. MR 3396210, DOI 10.1137/1.9781611972030.ch1
- T. J. R. Hughes, J. A. Cottrell, and Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Engrg. 194 (2005), no. 39-41, 4135–4195. MR 2152382, DOI 10.1016/j.cma.2004.10.008
- Lilia Krivodonova and Marsha Berger, High-order accurate implementation of solid wall boundary conditions in curved geometries, J. Comput. Phys. 211 (2006), no. 2, 492–512. MR 2173394, DOI 10.1016/j.jcp.2005.05.029
- M. Lenoir, Optimal isoparametric finite elements and error estimates for domains involving curved boundaries, SIAM J. Numer. Anal. 23 (1986), no. 3, 562–580. MR 842644, DOI 10.1137/0723036
- A. Main and G. Scovazzi, The shifted boundary method for embedded domain computations. Part I: Poisson and Stokes problems, J. Comput. Phys. 372 (2018), 972–995. MR 3847465, DOI 10.1016/j.jcp.2017.10.026
- 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, DOI 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, DOI 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, DOI 10.1137/S1064827503402837
- O. Zienkiewicz, R. Taylor, and R. Taylor, The Finite Element Method, McGraw-Hill, London, 1977.
Additional Information
- James Cheung
- Affiliation: Interdisciplinary Center for Applied Mathematics, Virginia Tech, Blacksburg, Virginia 24061
- MR Author ID: 1230885
- Email: jamescheung@vt.edu
- Mauro Perego
- Affiliation: Center for Computing Research, Sandia National Laboratories, Albuquerque, New Mexico 87123
- MR Author ID: 869976
- Email: mperego@sandia.gov
- Pavel Bochev
- Affiliation: Center for Computing Research, Sandia National Laboratories, Albuquerque, New Mexico 87123
- MR Author ID: 38390
- Email: pbboche@sandia.gov
- Max Gunzburger
- Affiliation: Department of Scientific Computing, Florida State University, Tallahassee, Florida 32309
- MR Author ID: 78360
- Email: mgunzburger@fsu.edu
- Received by editor(s): February 8, 2018
- Received by editor(s) in revised form: October 28, 2018
- Published electronically: February 21, 2019
- 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. - Journal: Math. Comp. 88 (2019), 2187-2219
- MSC (2010): Primary 65N30
- DOI: https://doi.org/10.1090/mcom/3415
- MathSciNet review: 3957891