Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T06:34:44.246Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical analysis of strongly nonlinear PDEs*

Published online by Cambridge University Press:  05 May 2017

Michael Neilan ,
Abner J. Salgado  and
Wujun Zhang
Michael Neilan
Affiliation:
Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, USA E-mail: neilan@pitt.edu
Abner J. Salgado
Affiliation:
Department of Mathematics, The University of Tennessee, Knoxville, TN, USA E-mail: asalgad1@utk.edu
Wujun Zhang
Affiliation:
Department of Mathematics, Rutgers University, Piscataway, NJ, USA E-mail: wujun@math.rutgers.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We review the construction and analysis of numerical methods for strongly nonlinear PDEs, with an emphasis on convex and non-convex fully nonlinear equations and the convergence to viscosity solutions. We begin by describing a fundamental result in this area which states that stable, consistent and monotone schemes converge as the discretization parameter tends to zero. We review methodologies to construct finite difference, finite element and semi-Lagrangian schemes that satisfy these criteria, and, in addition, discuss some rather novel tools that have paved the way to derive rates of convergence within this framework.

Type
Research Article
Information
Acta Numerica , Volume 26 , 01 May 2017 , pp. 137 - 303
Copyright
© Cambridge University Press, 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Alla, A., Falcone, M. and Kalise, D. (2015), ‘An efficient policy iteration algorithm for dynamic programming equations’, SIAM J. Sci. Comput. 37, A181–A200.Google Scholar
Awanou, G. (2015a), ‘Pseudo transient continuation and time marching methods for Monge–Ampère type equations’, Adv. Comput. Math. 41, 907–935.Google Scholar
Awanou, G. (2015b), ‘Quadratic mixed finite element approximations of the Monge–Ampère equation in 2D’, Calcolo 52, 503–518.CrossRefGoogle Scholar
Awanou, G. (2015c), ‘Standard finite elements for the numerical resolution of the elliptic Monge–Ampère equations: Classical solutions’, IMA J. Numer. Anal. 35, 1150–1166.CrossRefGoogle Scholar
Baiocchi, C. (1977), Estimations d’erreur dans L ∞ pour les inéquations à obstacle. In Mathematical Aspects of Finite Element Methods: Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.) Rome, 1975, Vol. 606 of Lecture Notes in Mathematics , Springer, pp. 27–34.Google Scholar
Barles, G. and Jakobsen, E. (2005), ‘Error bounds for monotone approximation schemes for Hamilton–Jacobi–Bellman equations’, SIAM J. Numer. Anal. 43, 540–558.Google Scholar
Barles, G. and Jakobsen, E. (2002), ‘On the convergence rate of approximation schemes for Hamilton–Jacobi–Bellman equations’, M2AN Math. Model. Numer. Anal. 36, 33–54.CrossRefGoogle Scholar
Barles, G. and Souganidis, P. (1991), ‘Convergence of approximation schemes for fully nonlinear second order equations’, Asymptotic Anal. 4, 271–283.Google Scholar
Bellman, R. (2010), Dynamic Programming, Princeton Landmarks in Mathematics, Princeton University Press. Reprint of the 1957 edition.Google Scholar
Benamou, J.-D. and Brenier, Y. (1998), ‘Weak existence for the semigeostrophic equations formulated as a coupled Monge–Ampère/transport problem’, SIAM J. Appl. Math. 58, 1450–1461.Google Scholar
Benamou, J.-D. and Brenier, Y. (2000), ‘A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem’, Numer. Math. 84, 375–393.CrossRefGoogle Scholar
Benamou, J.-D., Collino, F. and Mirebeau, J.-M. (2016), ‘Monotone and consistent discretization of the Monge–Ampère operator’, Math. Comp. 85(302), 2743–2775.Google Scholar
Benamou, J.-D., Froese, B. and Oberman, A. (2014), ‘Numerical solution of the optimal transportation problem using the Monge–Ampère equation’, J. Comput. Phys. 260, 107–126.Google Scholar
Bensoussan, A. and Lions, J.-L. (1984), Impulse Control and Quasivariational Inequalities. (translated from the French by J. M. Cole) Gauthier-Villars.Google Scholar
Böhmer, K. (2008), ‘On finite element methods for fully nonlinear elliptic equations of second order’, SIAM J. Numer. Anal. 46, 1212–1249.Google Scholar
Bokanowski, O., Maroso, S. and Zidani, H. (2009), ‘Some convergence results for Howard’s algorithm’, SIAM J. Numer. Anal. 47, 3001–3026.CrossRefGoogle Scholar
Boulbrachene, M. and Cortey-Dumont, P. (2009), ‘Optimal $L^{\infty }$ -error estimate of a finite element method for Hamilton–Jacobi–Bellman equations’, Numer. Funct. Anal. Optim. 30, 421–435.Google Scholar
Boulbrachene, M. and Haiour, M. (2001), ‘The finite element approximation of Hamilton–Jacobi–Bellman equations’, Comput. Math. Appl. 41, 993–1007.Google Scholar
Brenner, S. and Neilan, M. (2012), ‘Finite element approximations of the three dimensional Monge–Ampère equation’, ESAIM Math. Model. Numer. Anal. 46, 979–1001.Google Scholar
Brenner, S. and Scott, L. (2008), The Mathematical Theory of Finite Element Methods, third edition, Vol. 15 of Texts in Applied Mathematics , Springer.CrossRefGoogle Scholar
Brenner, S., Gudi, T., Neilan, M. and Sung, L.-Y. (2011), ‘ $C^{0}$ penalty methods for the fully nonlinear Monge–Ampère equation’, Math. Comp. 80(276), 1979–1995.Google Scholar
Caffarelli, L. and Cabré, X. (1995), Fully Nonlinear Elliptic Equations, Vol. 43 of American Mathematical Society Colloquium Publications , AMS.Google Scholar
Caffarelli, L. and Silvestre, L. (2010a), ‘On the Evans–Krylov theorem’, Proc. Amer. Math. Soc. 138, 263–265.Google Scholar
Caffarelli, L. and Silvestre, L. (2010b), Smooth approximations of solutions to nonconvex fully nonlinear elliptic equations. In Nonlinear Partial Differential Equations and Related Topics, Vol. 229 of American Mathematical Society Translations, Series 2, AMS, pp. 67–85.Google Scholar
Caffarelli, L. and Souganidis, P. (2008), ‘A rate of convergence for monotone finite difference approximations to fully nonlinear, uniformly elliptic PDEs’, Comm. Pure Appl. Math. 61, 1–17.CrossRefGoogle Scholar
Calderón, A. and Zygmund, A. (1952), ‘On the existence of certain singular integrals’, Acta Math. 88, 85–139.Google Scholar
Carnicer, J. and Dahmen, W. (1994), ‘Characterization of local strict convexity preserving interpolation methods by $C^{1}$ functions’, J. Approx. Theory 77, 2–30.Google Scholar
Ciarlet, P. (2002), The Finite Element Method for Elliptic Problems, Vol. 40 of Classics in Applied Mathematics , SIAM. Reprint of the 1978 original.Google Scholar
Cortey-Dumont, P. (1987), ‘Sur l’analyse numérique des équations de Hamilton–Jacobi–Bellman’, Math. Methods Appl. Sci. 9, 198–209.Google Scholar
Crandall, M. and Lions, P.-L. (1983), ‘Viscosity solutions of Hamilton–Jacobi equations’, Trans. Amer. Math. Soc. 277, 1–42.Google Scholar
Crandall, M., Ishii, H. and Lions, P.-L. (1992), ‘User’s guide to viscosity solutions of second order partial differential equations’, Bull. Amer. Math. Soc. (N.S.) 27, 1–67.Google Scholar
Dacorogna, B. (2008), Direct Methods in the Calculus of Variations, second edition, Vol. 78 of Applied Mathematical Sciences , Springer.Google Scholar
Dean, E. and Glowinski, R. (2006a), ‘An augmented Lagrangian approach to the numerical solution of the Dirichlet problem for the elliptic Monge–Ampère equation in two dimensions’, Electron. Trans. Numer. Anal. 22, 71–96.Google Scholar
Dean, E. and Glowinski, R. (2006b), ‘Numerical methods for fully nonlinear elliptic equations of the Monge–Ampère type’, Comput. Methods Appl. Mech. Engrg 195, 1344–1386.Google Scholar
Debrabant, K. and Jakobsen, E. (2013), ‘Semi-Lagrangian schemes for linear and fully non-linear diffusion equations’, Math. Comp. 82(283), 1433–1462.Google Scholar
Dong, H. and Krylov, N. (2007), ‘The rate of convergence of finite-difference approximations for parabolic Bellman equations with Lipschitz coefficients in cylindrical domains’, Appl. Math. Optim. 56, 37–66.CrossRefGoogle Scholar
Ekeland, I. and Témam, R. (1999), Convex Analysis and Variational Problems, Vol. 28 of Classics in Applied Mathematics , English edition, SIAM.Google Scholar
Ern, A. and Guermond, J.-L. (2004), Theory and Practice of Finite Elements, Vol. 159 of Applied Mathematical Sciences , Springer.Google Scholar
Evans, L. (1980), ‘On solving certain nonlinear partial differential equations by accretive operator methods’, Israel J. Math. 36, 225–247.Google Scholar
Evans, L. (2010), Partial Differential Equations, second edition, Vol. 19 of Graduate Studies in Mathematics , AMS.Google Scholar
Evans, L. and Friedman, A. (1979), ‘Optimal stochastic switching and the Dirichlet problem for the Bellman equation’, Trans. Amer. Math. Soc. 253, 365–389.CrossRefGoogle Scholar
Feng, X. and Jensen, M. (2017), ‘Convergent semi-Lagrangian methods for the Monge–Ampère equation on unstructured grids’, SIAM J. Numer. Anal., to appear.Google Scholar
Feng, X. and Lewis, T. (2014), ‘Local discontinuous Galerkin methods for one-dimensional second order fully nonlinear elliptic and parabolic equations’, J. Sci. Comput. 59, 129–157.Google Scholar
Feng, X. and Neilan, M. (2009), ‘Mixed finite element methods for the fully nonlinear Monge–Ampère equation based on the vanishing moment method’, SIAM J. Numer. Anal. 47, 1226–1250.CrossRefGoogle Scholar
Feng, X. and Neilan, M. (2011), ‘Analysis of Galerkin methods for the fully nonlinear Monge–Ampère equation’, J. Sci. Comput. 47, 303–327.Google Scholar
Feng, X., Glowinski, R. and Neilan, M. (2013), ‘Recent developments in numerical methods for fully nonlinear second order partial differential equations’, SIAM Rev. 55, 205–267.Google Scholar
Feng, X., Hennings, L. and Neilan, M. (2017), ‘ $C^{0}$ discontinuous Galerkin finite element methods for second order linear elliptic partial differential equations in non-divergence form’, Math. Comp., to appear.CrossRefGoogle Scholar
Feng, X., Neilan, M. and Schnake, S. (2016), Interior penalty discontinuous Galerkin methods for second order linear non-divergence form elliptic PDEs. arXiv:1605.04364 Google Scholar
Fleming, W. and Soner, H. (2006), Controlled Markov Processes and Viscosity Solutions, second edition, Vol. 25 of Stochastic Modelling and Applied Probability , Springer.Google Scholar
Fleming, W. and Souganidis, P. (1989), ‘On the existence of value functions of two-player, zero-sum stochastic differential games’, Indiana Univ. Math. J. 38, 293–314.CrossRefGoogle Scholar
Froese, B. (2016), Convergent approximation of surfaces of prescribed Gaussian curvature with weak Dirichlet conditions. arXiv:1601.06315 Google Scholar
Froese, B. and Oberman, A. (2011), ‘Convergent finite difference solvers for viscosity solutions of the elliptic Monge–Ampère equation in dimensions two and higher’, SIAM J. Numer. Anal. 49, 1692–1714.Google Scholar
Gallistl, D. (2017), ‘Variational formulation and numerical analysis of linear elliptic equations in nondivergence form with Cordes coefficients’, SIAM J. Numer. Anal., to appear.CrossRefGoogle Scholar
Georgoulis, E., Houston, P. and Virtanen, J. (2011), ‘An a posteriori error indicator for discontinuous Galerkin approximations of fourth-order elliptic problems’, IMA J. Numer. Anal. 31, 281–298.Google Scholar
Gerschgorin, S. (1930), ‘Fehlerabschätzung für das Differenzenverfahren zur Lösung partieller Differentialgleichungen’, Z. Angew. Math. Mech. 10, 373–382.Google Scholar
Gilbarg, D. and Trudinger, N. (2001), Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer. Reprint of the 1998 edition.Google Scholar
González, R. and Tidball, M. (1992), Fast solution of general nonlinear fixed point problems. In System Modelling and Optimization (Zurich, 1991), Vol. 180 of Lecture Notes in Control and Information Sciences , Springer, pp. 35–44.Google Scholar
Grisvard, P. (2011), Elliptic Problems in Nonsmooth Domains, Vol. 69 of Classics in Applied Mathematics , SIAM. Reprint of the 1985 original.Google Scholar
Gutiérrez, C. (2001), The Monge–Ampère Equation, Vol. 44 of Progress in Nonlinear Differential Equations and their Applications , Birkhä user.CrossRefGoogle Scholar
Han, D. and Wan, J. (2013), ‘Multigrid methods for second order Hamilton–Jacobi–Bellman and Hamilton–Jacobi–Bellman–Isaacs equations’, SIAM J. Sci. Comput. 35, S323–S344.Google Scholar
Han, Q. and Lin, F. (2011), second edition, Vol. 1 of Courant Lecture Notes in Mathematics , AMS.Google Scholar
Hintermüller, M., Ito, K. and Kunisch, K. (2002), ‘The primal–dual active set strategy as a semismooth Newton method’, SIAM J. Optim. 13, 865–888.Google Scholar
Holden, H. and Risebro, N. (2015), Front Tracking for Hyperbolic Conservation Laws, second edition, Vol. 152 of Applied Mathematical Sciences , Springer.Google Scholar
Hoppe, R. (1986), ‘Multigrid methods for Hamilton–Jacobi–Bellman equations’, Numer. Math. 49, 239–254.CrossRefGoogle Scholar
Iserles, A. (2009), A First Course in the Numerical Analysis of Differential Equations, second edition, Cambridge Texts in Applied Mathematics, Cambridge University Press.Google Scholar
Ishii, H. (1995), ‘On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions’, Funkcial. Ekvac. 38, 101–120.Google Scholar
Jakobsen, E. (2004), ‘On error bounds for approximation schemes for non-convex degenerate elliptic equations’, BIT 44, 269–285.Google Scholar
Jakobsen, E. (2006), ‘On error bounds for monotone approximation schemes for multi-dimensional Isaacs equations’, Asymptot. Anal. 49, 249–273.Google Scholar
Jensen, M. and Smears, I. (2013a), Finite element methods with artificial diffusion for Hamilton–Jacobi–Bellman equations. In Numerical Mathematics and Advanced Applications 2011 (Cangiani, A. et al. , ed.), Springer, pp. 267–274.Google Scholar
Jensen, M. and Smears, I. (2013b), ‘On the convergence of finite element methods for Hamilton–Jacobi–Bellman equations’, SIAM J. Numer. Anal. 51, 137–162.Google Scholar
Katsoulakis, M. (1995), ‘A representation formula and regularizing properties for viscosity solutions of second-order fully nonlinear degenerate parabolic equations’, Nonlinear Anal. 24, 147–158.Google Scholar
Katzourakis, N. (2015), An Introduction to Viscosity Solutions for Fully Nonlinear PDE with Applications to Calculus of Variations in L ∞ , Springer Briefs in Mathematics, Springer.Google Scholar
Kawohl, B. and Kutev, N. (2007), ‘Comparison principle for viscosity solutions of fully nonlinear, degenerate elliptic equations’, Comm. Partial Diff. Equations 32, 1209–1224.Google Scholar
Kocan, M. (1995), ‘Approximation of viscosity solutions of elliptic partial differential equations on minimal grids’, Numer. Math. 72, 73–92.Google Scholar
Koike, S. (2004), A Beginner’s Guide to the Theory of Viscosity Solutions, Vol. 13 of MSJ Memoirs , Mathematical Society of Japan.Google Scholar
Koike, S. and Świech, A. (2013), ‘Representation formulas for solutions of Isaacs integro-PDE’, Indiana Univ. Math. J. 62, 1473–1502.Google Scholar
Kossaczký, I., Ehrhardt, M. and Günther, M. (2016), ‘On the non-existence of higher order monotone approximation schemes for HJB equations’, Appl. Math. Lett. 52, 53–57.Google Scholar
Krylov, N. (1987), Nonlinear Elliptic and Parabolic Equations of the Second Order (translated from the Russian by P. L. Buzytsky), Vol. 7 of Mathematics and its Applications (Soviet Series) , Reidel.Google Scholar
Krylov, N. (1996), Lectures on Elliptic and Parabolic Equations in Hölder Spaces, Vol. 12 of Graduate Studies in Mathematics , AMS.Google Scholar
Krylov, N. (1997), ‘On the rate of convergence of finite-difference approximations for Bellman’s equations’, Algebra i Analiz 9, 245–256.Google Scholar
Krylov, N. (2000), ‘On the rate of convergence of finite-difference approximations for Bellman’s equations with variable coefficients’, Probab. Theory Rel. Fields 117, 1–16.Google Scholar
Krylov, N. (2005), ‘The rate of convergence of finite-difference approximations for Bellman equations with Lipschitz coefficients’, Appl. Math. Optim. 52, 365–399.Google Scholar
Krylov, N. (2008), Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, Vol. 96 of Graduate Studies in Mathematics , AMS.Google Scholar
Krylov, N. (2014), ‘On the rate of convergence of difference approximations for uniformly nondegenerate elliptic Bellman’s equations’, Appl. Math. Optim. 69, 431–458.Google Scholar
Krylov, N. (2015), ‘On the rate of convergence of finite-difference approximations for elliptic Isaacs equations in smooth domains’, Comm. Partial Diff. Equations 40, 1393–1407.Google Scholar
Kuo, H.-J. and Trudinger, N. (1990), ‘Linear elliptic difference inequalities with random coefficients’, Math. Comp. 55(191), 37–53.Google Scholar
Kuo, H.-J. and Trudinger, N. (1992), ‘Discrete methods for fully nonlinear elliptic equations’, SIAM J. Numer. Anal. 29, 123–135.Google Scholar
Kuo, H.-J. and Trudinger, N. (1996), ‘Positive difference operators on general meshes’, Duke Math. J. 83, 415–433.Google Scholar
Kuo, H.-J. and Trudinger, N. (2000), ‘A note on the discrete Aleksandrov–Bakelman maximum principle’, Taiwanese J. Math. 4, 55–64.Google Scholar
Lai, M.-J. and Schumaker, L. (2007), ‘Trivariate $C^{r}$ polynomial macroelements’, Constr. Approx. 26, 11–28.Google Scholar
Lakkis, O. and Pryer, T. (2011), ‘A finite element method for second order nonvariational elliptic problems’, SIAM J. Sci. Comput. 33, 786–801.Google Scholar
Lions, P.-L. (1982), Generalized Solutions of Hamilton–Jacobi Equations, Vol. 69 of Research Notes in Mathematics , Pitman.Google Scholar
Lions, P.-L. (1983), ‘Optimal control of diffusion processes and Hamilton–Jacobi–Bellman equations II: Viscosity solutions and uniqueness’, Comm. Partial Diff. Equations 8, 1229–1276.Google Scholar
Lions, P.-L. and Mercier, B. (1980), ‘Approximation numérique des équations de Hamilton–Jacobi–Bellman’, RAIRO Anal. Numér. 14, 369–393.Google Scholar
Maugeri, A., Palagachev, D. and Softova, L. (2000), Elliptic and Parabolic Equations with Discontinuous Coefficients, Vol. 109 of Mathematical Research , Wiley–VCH.Google Scholar
Menaldi, J.-L. (1989), ‘Some estimates for finite difference approximations’, SIAM J. Control Optim. 27, 579–607.Google Scholar
Mingione, G. (2006), ‘Regularity of minima: An invitation to the dark side of the calculus of variations’, Appl. Math. 51, 355–426.Google Scholar
Mirebeau, J.-M. (2015), ‘Discretization of the 3D Monge–Ampere operator, between wide stencils and power diagrams’, ESAIM Math. Model. Numer. Anal. 49, 1511–1523.Google Scholar
Motzkin, T. and Wasow, W. (1953), ‘On the approximation of linear elliptic differential equations by difference equations with positive coefficients’, J. Math. Phys. 31, 253–259.Google Scholar
Nadirashvili, N. and Vlăduţ, S. (2007), ‘Nonclassical solutions of fully nonlinear elliptic equations’, Geom. Funct. Anal. 17, 1283–1296.Google Scholar
Nadirashvili, N. and Vlăduţ, S. (2008), ‘Singular viscosity solutions to fully nonlinear elliptic equations’, J. Math. Pures Appl. (9) 89, 107–113.Google Scholar
Nadirashvili, N., Tkachev, V. and Vlăduţ, S. (2012), ‘A non-classical solution to a Hessian equation from Cartan isoparametric cubic’, Adv. Math. 231, 1589–1597.Google Scholar
Neilan, M. (2010), ‘A nonconforming Morley finite element method for the fully nonlinear Monge–Ampère equation’, Numer. Math. 115, 371–394.Google Scholar
Neilan, M. (2013), ‘Quadratic finite element approximations of the Monge–Ampère equation’, J. Sci. Comput. 54, 200–226.Google Scholar
Neilan, M. (2017), ‘Convergence analysis of a finite element method for second order non-variational elliptic problems’, J. Numer. Math., to appear.Google Scholar
Nirenberg, L. (1953), ‘On nonlinear elliptic partial differential equations and Hölder continuity’, Comm. Pure Appl. Math. 6, 103–156; addendum, 395.CrossRefGoogle Scholar
Nochetto, R. and Zhang, W. (2017a), ‘Discrete ABP estimate and convergence rates for linear elliptic equations in non-divergence form’, Found. Comput. Math., to appear.Google Scholar
Nochetto, R. and Zhang, W. (2017b), Pointwise rates of convergence for the Oliker–Prussner’s method for the Monge–Ampère equation. arXiv:1611.02786 Google Scholar
Nochetto, R., Ntogakas, D. and Zhang, W. (2017), Two-scale method for the Monge–Ampère equation: Convergence rates. Preprint.Google Scholar
Nochetto, R., Otárola, E. and Salgado, A. J. (2015), ‘Convergence rates for the classical, thin and fractional elliptic obstacle problems’, Philos. Trans. A 373(2050), 20140449.Google Scholar
Oberman, A. (2006), ‘Convergent difference schemes for degenerate elliptic and parabolic equations: Hamilton–Jacobi equations and free boundary problems’, SIAM J. Numer. Anal. 44, 879–895.Google Scholar
Oberman, A. (2008), ‘Wide stencil finite difference schemes for the elliptic Monge–Ampère equation and functions of the eigenvalues of the Hessian’, Discrete Contin. Dyn. Syst. Ser. B 10, 221–238.Google Scholar
Øksendal, B. (2003), Stochastic Differential Equations: An Introduction with Applications, sixth edition, Universitext, Springer.Google Scholar
Oliker, V. and Prussner, L. (1988), ‘On the numerical solution of the equation $(\unicode[STIX]{x2202}^{2}z/\unicode[STIX]{x2202}x^{2})(\unicode[STIX]{x2202}^{2}z/\unicode[STIX]{x2202}y^{2})-((\unicode[STIX]{x2202}^{2}z/\unicode[STIX]{x2202}x\unicode[STIX]{x2202}y))^{2}=f$ and its discretizations, I’, Numer. Math. 54, 271–293.Google Scholar
Safonov, M. (1980), ‘Harnack’s inequality for elliptic equations and Hölder property of their solutions’, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 96, 272–287; 312.Google Scholar
Safonov, M. (1987), ‘Unimprovability of estimates of Hölder constants for solutions of linear elliptic equations with measurable coefficients’, Mat. Sb. (N.S.) 132(174), 275–288.Google Scholar
Salgado, A. J. and Zhang, W. (2016), Finite element approximation of the Isaacs equation. arXiv:1512.09091v1 Google Scholar
Schatz, A. and Wahlbin, L. (1982), ‘On the quasi-optimality in $L_{\infty }$ of the ${\dot{H}}^{1}$ -projection into finite element spaces’, Math. Comp. 38(157), 1–22.Google Scholar
Silvestre, L. (2015), Viscosity solutions of elliptic equations. http://math.uchicago.edu/∼luis/preprints/viscosity-solutions.pdf Google Scholar
Smears, I. and Süli, E. (2013), ‘Discontinuous Galerkin finite element approximation of nondivergence form elliptic equations with Cordès coefficients’, SIAM J. Numer. Anal. 51, 2088–2106.Google Scholar
Smears, I. and Süli, E. (2014), ‘Discontinuous Galerkin finite element approximation of Hamilton–Jacobi–Bellman equations with Cordes coefficients’, SIAM J. Numer. Anal. 52, 993–1016.Google Scholar
Smears, I. and Süli, E. (2016), ‘Discontinuous Galerkin finite element methods for time-dependent Hamilton–Jacobi–Bellman equations with Cordes coefficients’, Numer. Math. 133, 141–176.Google Scholar
Stojanovic, S. (2005), ‘Risk premium and fair option prices under stochastic volatility: The HARA solution’, CR Math. Acad. Sci. Paris 340, 551–556.Google Scholar
Strang, G. and Fix, G. (1973), An Analysis of the Finite Element Method Prentice-Hall Series in Automatic Computation, Prentice-Hall.Google Scholar
Tolstoy, L. (1998), Anna Karenina, Project Gutenberg, EBook #1399.Google Scholar
Trudinger, N. (1988), ‘Comparison principles and pointwise estimates for viscosity solutions of nonlinear elliptic equations’, Rev. Mat. Iberoamericana 4, 453–468.Google Scholar
Trudinger, N. and Wang, X.-J. (2005), ‘The affine Plateau problem’, J. Amer. Math. Soc. 18, 253–289.Google Scholar
Trudinger, N. and Wang, X.-J. (2008), ‘Boundary regularity for the Monge–Ampère and affine maximal surface equations’, Ann. of Math. (2) 167, 993–1028.Google Scholar
Turanova, O. (2015), ‘Error estimates for approximations of nonhomogeneous nonlinear uniformly elliptic equations’, Calc. Var. Partial Diff. Equations 54, 2939–2983.Google Scholar
Villani, C. (2003), Topics in Optimal Transportation, Vol. 58 of Graduate Studies in Mathematics , AMS.Google Scholar
Wang, C. and Wang, J. (2017), ‘A primal–dual weak Galerkin finite element method for second order elliptic equations in non-divergence form’, Math. Comp., to appear.CrossRefGoogle Scholar
Wang, X.-J. (1996), ‘Regularity for Monge–Ampère equation near the boundary’, Analysis 16, 101–107.Google Scholar
Witte, J. and Reisinger, C. (2012), ‘Penalty methods for the solution of discrete HJB equations: Continuous control and obstacle problems’, SIAM J. Numer. Anal. 50, 595–625.Google Scholar
Xu, J. and Zikatanov, L. (1999), ‘A monotone finite element scheme for convection-diffusion equations’, Math. Comp. 68(228), 1429–1446.Google Scholar
Ženíšek, A. (1973), Hermite interpolation on simplexes in the finite element method. In Proc. Equadiff III: Third Czechoslovak Conf. Differential Equations and their Appl., Brno, 1972, Purkyně University, Brno, 271–277. Vol. 1 of Folia Fac. Sci. Natur. Univ. Purkynianae Brunensis, Ser. Monograph .Google Scholar