Numerical approximation of planar oblique derivative problems in nondivergence form
HTML articles powered by AMS MathViewer
- by Dietmar Gallistl HTML | PDF
- Math. Comp. 88 (2019), 1091-1119 Request permission
Abstract:
A numerical method for approximating a uniformly elliptic oblique derivative problem in two-dimensional simply-connected domains is proposed. The numerical scheme employs a mixed formulation with piecewise affine functions on curved finite element domains. The direct approximation of the gradient of the solution turns the oblique derivative boundary condition into an oblique direction condition. A priori and a posteriori error estimates as well as numerical computations on uniform and adaptive meshes are provided.References
- John W. Barrett and Charles M. Elliott, Fixed mesh finite element approximations to a free boundary problem for an elliptic equation with an oblique derivative boundary condition, Comput. Math. Appl. 11 (1985), no. 4, 335–345. MR 789678, DOI 10.1016/0898-1221(85)90058-6
- Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275
- Daniele Boffi, Franco Brezzi, and Michel Fortin, Mixed finite element methods and applications, Springer Series in Computational Mathematics, vol. 44, Springer, Heidelberg, 2013. MR 3097958, DOI 10.1007/978-3-642-36519-5
- Dietrich Braess, Finite elements, 3rd ed., Cambridge University Press, Cambridge, 2007. Theory, fast solvers, and applications in elasticity theory; Translated from the German by Larry L. Schumaker. MR 2322235, DOI 10.1017/CBO9780511618635
- 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
- Sunčica Čanić, Barbara Lee Keyfitz, and Gary M. Lieberman, A proof of existence of perturbed steady transonic shocks via a free boundary problem, Comm. Pure Appl. Math. 53 (2000), no. 4, 484–511. MR 1733695, DOI 10.1002/(SICI)1097-0312(200004)53:4<484::AID-CPA3>3.3.CO;2-B
- Ph. Clément, Approximation by finite element functions using local regularization, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. 9 (1975), no. R-2, 77–84 (English, with Loose French summary). MR 0400739
- Daniele Antonio Di Pietro and Alexandre Ern, Mathematical aspects of discontinuous Galerkin methods, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 69, Springer, Heidelberg, 2012. MR 2882148, DOI 10.1007/978-3-642-22980-0
- Willy Dörfler, A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal. 33 (1996), no. 3, 1106–1124. MR 1393904, DOI 10.1137/0733054
- Z. Fašková, R. Čunderlík, and K. Mikula, Finite element method for solving geodetic boundary value problems, Journal of Geodesy 84 (2010), no. 2, 135–144.
- Xiaobing Feng, Lauren Hennings, and Michael Neilan, Finite element methods for second order linear elliptic partial differential equations in non-divergence form, Math. Comp. 86 (2017), no. 307, 2025–2051. MR 3647950, DOI 10.1090/mcom/3168
- Xiaobing Feng and Max Jensen, Convergent semi-Lagrangian methods for the Monge-Ampère equation on unstructured grids, SIAM J. Numer. Anal. 55 (2017), no. 2, 691–712. MR 3623696, DOI 10.1137/16M1061709
- Xiaobing Feng, Michael Neilan, and Stefan Schnake, Interior penalty discontinuous Galerkin methods for second order linear non-divergence form elliptic PDEs, J. Sci. Comput. 74 (2018), no. 3, 1651–1676. MR 3767822, DOI 10.1007/s10915-017-0519-3
- Dietmar Gallistl, Stable splitting of polyharmonic operators by generalized Stokes systems, Math. Comp. 86 (2017), no. 308, 2555–2577. MR 3667017, DOI 10.1090/mcom/3208
- Dietmar Gallistl, Variational formulation and numerical analysis of linear elliptic equations in nondivergence form with Cordes coefficients, SIAM J. Numer. Anal. 55 (2017), no. 2, 737–757. MR 3628316, DOI 10.1137/16M1080495
- D. Gallistl, Mixed finite element approximation of elliptic equations involving high-order derivatives, Habilitation thesis, Karlsruher Institut für Technologie, Fakultät für Mathematik, 2018.
- E. Kawecki, A DGFEM for uniformly elliptic two dimensional oblique boundary value problems, arXiv e-prints 1711.01836.
- N. V. Krylov, Nonlinear elliptic and parabolic equations of the second order, Mathematics and its Applications (Soviet Series), vol. 7, D. Reidel Publishing Co., Dordrecht, 1987. Translated from the Russian by P. L. Buzytsky [P. L. Buzytskiĭ]. MR 901759, DOI 10.1007/978-94-010-9557-0
- Omar Lakkis and Tristan Pryer, A finite element method for second order nonvariational elliptic problems, SIAM J. Sci. Comput. 33 (2011), no. 2, 786–801. MR 2801189, DOI 10.1137/100787672
- Gary M. Lieberman, Oblique derivative problems for elliptic equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. MR 3059278, DOI 10.1142/8679
- P.-L. Lions, Two remarks on Monge-Ampère equations, Ann. Mat. Pura Appl. (4) 142 (1985), 263–275 (1986). MR 839040, DOI 10.1007/BF01766596
- Antonino Maugeri, Dian K. Palagachev, and Lubomira G. Softova, Elliptic and parabolic equations with discontinuous coefficients, Mathematical Research, vol. 109, Wiley-VCH Verlag Berlin GmbH, Berlin, 2000. MR 2260015, DOI 10.1002/3527600868
- Matej Medl’a, Karol Mikula, Róbert Čunderlík, and Marek Macák, Numerical solution to the oblique derivative boundary value problem on non-uniform grids above the Earth topography, Journal of Geodesy 92 (2018), no. 1, 1–19.
- Ricardo H. Nochetto and Wujun Zhang, Discrete ABP estimate and convergence rates for linear elliptic equations in non-divergence form, Found. Comput. Math. 18 (2018), no. 3, 537–593. MR 3807356, DOI 10.1007/s10208-017-9347-y
- Peter Oswald, Multilevel finite element approximation, Teubner Skripten zur Numerik. [Teubner Scripts on Numerical Mathematics], B. G. Teubner, Stuttgart, 1994. Theory and applications. MR 1312165, DOI 10.1007/978-3-322-91215-2
- Dian K. Palagachev, The Poincaré problem in $L^p$-Sobolev spaces. II. Full dimension degeneracy, Comm. Partial Differential Equations 33 (2008), no. 1-3, 209–234. MR 2398226, DOI 10.1080/03605300701454933
- Mikhail V. Safonov, Nonuniqueness for second-order elliptic equations with measurable coefficients, SIAM J. Math. Anal. 30 (1999), no. 4, 879–895. MR 1684729, DOI 10.1137/S0036141096309046
- Ridgway Scott, Interpolated boundary conditions in the finite element method, SIAM J. Numer. Anal. 12 (1975), 404–427. MR 386304, DOI 10.1137/0712032
- Iain Smears and Endre Süli, Discontinuous Galerkin finite element approximation of nondivergence form elliptic equations with Cordès coefficients, SIAM J. Numer. Anal. 51 (2013), no. 4, 2088–2106. MR 3077903, DOI 10.1137/120899613
- Iain Smears and Endre Süli, Discontinuous Galerkin finite element approximation of Hamilton-Jacobi-Bellman equations with Cordes coefficients, SIAM J. Numer. Anal. 52 (2014), no. 2, 993–1016. MR 3196952, DOI 10.1137/130909536
- Iain Smears and Endre Süli, Discontinuous Galerkin finite element methods for time-dependent Hamilton-Jacobi-Bellman equations with Cordes coefficients, Numer. Math. 133 (2016), no. 1, 141–176. MR 3475658, DOI 10.1007/s00211-015-0741-6
- Giorgio Talenti, Problemi di derivata obliqua per equazioni ellittiche in due variabili, Boll. Un. Mat. Ital. (3) 22 (1967), 505–526 (Italian). MR 0231048
- John Urbas, On the second boundary value problem for equations of Monge-Ampère type, J. Reine Angew. Math. 487 (1997), 115–124. MR 1454261, DOI 10.1515/crll.1997.487.115
- Rüdiger Verfürth, A posteriori error estimation techniques for finite element methods, Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford, 2013. MR 3059294, DOI 10.1093/acprof:oso/9780199679423.001.0001
- Yuxi Zheng, A global solution to a two-dimensional Riemann problem involving shocks as free boundaries, Acta Math. Appl. Sin. Engl. Ser. 19 (2003), no. 4, 559–572. MR 2019399, DOI 10.1007/210255-003-0131-1
Additional Information
- Dietmar Gallistl
- Affiliation: Department of Applied Mathematics, University of Twente, 7500 AE Enschede, The Netherlands
- MR Author ID: 1020312
- Email: d.gallistl@utwente.nl
- Received by editor(s): November 28, 2017
- Received by editor(s) in revised form: February 27, 2018, and April 15, 2018
- Published electronically: July 23, 2018
- Additional Notes: The author was supported by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173.
- © Copyright 2018 American Mathematical Society
- Journal: Math. Comp. 88 (2019), 1091-1119
- MSC (2010): Primary 65N12, 65N15, 65N30
- DOI: https://doi.org/10.1090/mcom/3371
- MathSciNet review: 3904140