This paper is concerned with C0 (non-Lagrange) finite element approximations of the linear elliptic equations in non-divergence form and the Hamilton–Jacobi–Bellman (HJB) equations with Cordes coefficients. Motivated by the Miranda–Talenti estimate, a discrete analog is proved once the finite element space is C0 on the \((n-1)\)-dimensional subsimplex (face) and \(C^1\) on \((n-2)\)-dimensional subsimplex. The main novelty of the non-standard finite element methods is to introduce an interior stabilization term to argument the PDE-induced variational form of the linear elliptic equations in non-divergence form or the HJB equations. As a distinctive feature of the proposed methods, no stabilization parameter is involved in the variational forms. As a consequence, the coercivity constant (resp. monotonicity constant) for the linear elliptic equations in non-divergence form (resp. the HJB equations) at discrete level is exactly the same as that from PDE theory. The quasi-optimal order error estimates as well as the convergence of the semismooth Newton method are established. Numerical experiments are provided to validate the convergence theory and to illustrate the accuracy and computational efficiency of the proposed methods.
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Barles, G., Souganidis, P.E.: Convergence of approximation schemes for fully nonlinear second order equations. Asymptot. Anal. 4(3), 271–283 (1991)
Bonnans, J.F., Zidani, H.: Consistency of generalized finite difference schemes for the stochastic HJB equation. SIAM J. Numer. Anal. 41(3), 1008–1021 (2003)
Brenner, S., Scott, R.: The Mathematical Theory of Finite Element Methods, vol. 15. Springer, Berlin (2007)
Brenner, S.C., Kawecki, E.L.: Adaptive \(C^0\) interior penalty methods for Hamilton–Jacobi–Bellman equations with Cordes coefficients. J. Comput. Appl. Math. 388, 113241 (2020)
Caffarelli, L.A., Gutiérrez, C.E.: Properties of the solutions of the linearized Monge–Ampère equation. Am. J. Math. 119(2), 423–465 (1997)
Camilli, F., Jakobsen, E.R.: A finite element like scheme for integro-partial differential Hamilton–Jacobi–Bellman equations. SIAM J. Numer. Anal. 47(4), 2407–2431 (2009)
Christiansen, S.H., Hu, J., Hu, K.: Nodal finite element de Rham complexes. Numer. Math. 139(2), 411–446 (2018)
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)
Falk, R.S., Neilan, M.: Stokes complexes and the construction of stable finite elements with pointwise mass conservation. SIAM J. Numer. Anal. 51(2), 1308–1326 (2013)
Feng, X., Glowinski, R., Neilan, M.: Recent developments in numerical methods for fully nonlinear second order partial differential equations. SIAM Rev. 55(2), 205–267 (2013)
Feng, X., Hennings, L., Neilan, M.: Finite element methods for second order linear elliptic partial differential equations in non-divergence form. Math. Comput. 86(307), 2025–2051 (2017)
Feng, X., Lewis, T.: A narrow-stencil finite difference method for approximating viscosity solutions of fully nonlinear elliptic partial differential equations with applications to Hamilton–Jacobi–Bellman equations. arXiv preprint arXiv:1907.10204 (2019)
Feng, X., Neilan, M., Schnake, S.: Interior penalty discontinuous Galerkin methods for second order linear non-divergence form elliptic PDEs. J. Sci. Comput. 74(3), 1651–1676 (2018)
Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions, vol. 25. Springer, Berlin (2006)
Gallistl, D.: Variational formulation and numerical analysis of linear elliptic equations in nondivergence form with Cordes coefficients. SIAM J. Numer. Anal. 55(2), 737–757 (2017)
Gallistl, D.: Numerical approximation of planar oblique derivative problems in nondivergence form. Math. Comput. 88(317), 1091–1119 (2019)
Gallistl, D., Süli, E.: Mixed finite element approximation of the Hamilton–Jacobi–Bellman equation with Cordes coefficients. SIAM J. Numer. Anal. 57(2), 592–614 (2019)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2015)
Grisvard, P.: Elliptic Problems in Nonsmooth Domains. SIAM, Philadelphia (2011)
Jensen, M.: \(L^2(H_\gamma ^1)\) finite element convergence for degenerate isotropic Hamilton–Jacobi–Bellman equations. IMA J. Numer. Anal. 37(3), 1300–1316 (2017)
Jensen, M., Smears, I.: On the convergence of finite element methods for Hamilton–Jacobi–Bellman equations. SIAM J. Numer. Anal. 51(1), 137–162 (2013)
Kawecki, E.L.: A DGFEM for nondivergence form elliptic equations with Cordes coefficients on curved domains. Numer. Methods Partial Differ. Equ. 35(5), 1717–1744 (2019)
Kawecki, E.L.: A discontinuous Galerkin finite element method for uniformly elliptic two dimensional oblique boundary-value problems. SIAM J. Numer. Anal. 57(2), 751–778 (2019)
Kawecki, E.L., Smears, I.: Convergence of adaptive discontinuous Galerkin and \(C^0\)-interior penalty finite element methods for Hamilton–Jacobi–Bellman and Isaacs equations. arXiv preprint arXiv:2006.07215 (2020)
Kawecki, E.L., Smears, I.: Unified analysis of discontinuous Galerkin and \(C^0\)-interior penalty finite element methods for Hamilton–Jacobi–Bellman and Isaacs equations. arXiv preprint arXiv:2006.07202 (2020)
Kuratowski, K., Ryll-Nardzewski, C.: A general theorem on selectors. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 13(6), 397–403 (1965)
Lakkis, O., Pryer, T.: A finite element method for second order nonvariational elliptic problems. SIAM J. Sci. Comput. 33(2), 786–801 (2011)
Lakkis, O., Pryer, T.: A finite element method for nonlinear elliptic problems. SIAM J. Sci. Comput. 35(4), A2025–A2045 (2013)
Li, R., Yang, F.: A sequential least squares method for elliptic equations in non-divergence form. arXiv preprint arXiv:1906.03754 (2019)
Maugeri, A., Palagachev, D.K., Softova, L.G.: Elliptic and Parabolic Equations with Discontinuous Coefficients, vol. 109. Wiley, Weinheim (2000)
Neilan, M.: Discrete and conforming smooth de Rham complexes in three dimensions. Math. Comput. 84(295), 2059–2081 (2015)
Neilan, M., Salgado, A.J., Zhang, W.: Numerical analysis of strongly nonlinear PDEs. Acta Numer. 26, 137–303 (2017)
Neilan, M., Wu, M.: Discrete Miranda–Talenti estimates and applications to linear and nonlinear PDEs. J. Comput. Appl. Math. 356, 358–376 (2019)
Nochetto, R.H., Zhang, W.: Discrete ABP estimate and convergence rates for linear elliptic equations in non-divergence form. Found. Comput. Math. 18(3), 537–593 (2018)
Qiu, W., Zhang, S.: Adaptive first-order system least-squares finite element methods for second order elliptic equations in non-divergence form. arXiv preprint arXiv:1906.11436 (2019)
Renardy, M., Rogers, R.C.: An Introduction to Partial Differential Equations, vol. 13. Springer, Berlin (2006)
Salgado, A.J., Zhang, W.: Finite element approximation of the Isaacs equation. ESAIM Math. Model. Numer. Anal. 53(2), 351–374 (2019)
Smears, I., Süli, E.: Discontinuous Galerkin finite element approximation of nondivergence form elliptic equations with Cordes coefficients. SIAM J. Numer. Anal. 51(4), 2088–2106 (2013)
Smears, I., Süli, E.: Discontinuous Galerkin finite element approximation of Hamilton–Jacobi–Bellman equations with Cordes coefficients. SIAM J. Numer. Anal. 52(2), 993–1016 (2014)
Smears, I., Süli, E.: Discontinuous Galerkin finite element methods for time-dependent Hamilton–Jacobi–Bellman equations with Cordes coefficients. Numer. Math. 133(1), 141–176 (2016)
Ulbrich, M.: Semismooth Newton methods for operator equations in function spaces. SIAM J. Optim. 13(3), 805–841 (2002)
Wang, C., Wang, J.: A primal–dual weak Galerkin finite element method for second order elliptic equations in non-divergence form. Math. Comput. 87(310), 515–545 (2018)
The work of Shuonan Wu is supported in part by the National Natural Science Foundation of China Grant No. 11901016 and the startup grant from Peking University. The author would like to express his gratitude to Guangwei Gao and Prof. Jun Hu in Peking University for their helpful discussions, and to the anonymous referees for their valuable comments.
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Wu, S. C0 finite element approximations of linear elliptic equations in non-divergence form and Hamilton–Jacobi–Bellman equations with Cordes coefficients. Calcolo 58, 9 (2021). https://doi.org/10.1007/s10092-021-00400-1
- Elliptic PDEs in non-divergence form
- Hamilton–Jacobi–Bellman equations
- Cordes condition
- C 0 (non-Lagrange) finite element methods
Mathematics Subject Classification