{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,15]],"date-time":"2026-07-15T05:59:52Z","timestamp":1784095192984,"version":"3.55.0"},"reference-count":51,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2017,3,1]],"date-time":"2017-03-01T00:00:00Z","timestamp":1488326400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Found Comput Math"],"published-print":{"date-parts":[[2018,6]]},"DOI":"10.1007\/s10208-017-9347-y","type":"journal-article","created":{"date-parts":[[2017,3,1]],"date-time":"2017-03-01T20:25:16Z","timestamp":1488399916000},"page":"537-593","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":28,"title":["Discrete ABP Estimate and Convergence Rates for Linear Elliptic Equations in Non-divergence Form"],"prefix":"10.1007","volume":"18","author":[{"given":"Ricardo H.","family":"Nochetto","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wujun","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2017,3,1]]},"reference":[{"key":"9347_CR1","doi-asserted-by":"crossref","unstructured":"G.\u00a0Barles and E.\u00a0R. Jakobsen. Error bounds for monotone approximation schemes for Hamilton-Jacobi-Bellman equations. SIAM J. Numer. Anal., 43(2):540\u2013558 (electronic), 2005.","DOI":"10.1137\/S003614290343815X"},{"key":"9347_CR2","doi-asserted-by":"crossref","first-page":"271","DOI":"10.3233\/ASY-1991-4305","volume":"4","author":"G Barles","year":"1991","unstructured":"G.\u00a0Barles and P.\u00a0E. Souganidis. Convergence of approximation schemes for fully nonlinear second order equations. Asymptotic Anal., 4:271\u2013283, 1991.","journal-title":"Asymptotic Anal."},{"key":"9347_CR3","doi-asserted-by":"crossref","unstructured":"S. Bartels. Stability and convergence of finite-element approximation schemes for harmonic maps. SIAM J. Numer. Anal., 43(1):220\u2013238 (electronic), 2005.","DOI":"10.1137\/040606594"},{"key":"9347_CR4","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1016\/j.jcp.2013.12.015","volume":"260","author":"J-D Benamou","year":"2014","unstructured":"J.-D.\u00a0Benamou, B.\u00a0D.\u00a0Froese and A.\u00a0M. Oberman. Numerical solution of the optimal transportation problem using the Monge-Amp\u00e8re equation. J. Comput. Phys., 260:107\u2013126, 2014.","journal-title":"J. Comput. Phys."},{"key":"9347_CR5","doi-asserted-by":"crossref","unstructured":"S.\u00a0N. Bernstein. Sur la g\u00e9n\u00e9ralisation du probl\u00e9me de Dirichlet. Math. Ann., 62:253\u2013271, 1906; 69:82\u2013136. 1910.","DOI":"10.1007\/BF01449980"},{"issue":"3","key":"9347_CR6","doi-asserted-by":"crossref","first-page":"1008","DOI":"10.1137\/S0036142901387336","volume":"41","author":"JF Bonnans","year":"2003","unstructured":"J.\u00a0F. Bonnans and H.\u00a0Zidani. Consistency of generalized finite difference schemes for the stochastic HJB equation. SIAM J. Numer. Anal., 41(3):1008\u20131021, 2003.","journal-title":"SIAM J. Numer. Anal."},{"key":"9347_CR7","unstructured":"S. Brenner and L.R. Scott. The Mathematical Theory of Finite Element Methods. Berlin: Springer 2014."},{"issue":"253\u2013284","key":"9347_CR8","first-page":"1988","volume":"58","author":"L Caffarelli","year":"1990","unstructured":"L.\u00a0Caffarelli. Elliptic second order equations. Rend. Sem. Mat. Fis. Milano, 58:253\u2013284 (1990), 1988.","journal-title":"Rend. Sem. Mat. Fis. Milano"},{"key":"9347_CR9","doi-asserted-by":"crossref","unstructured":"L.\u00a0Caffarelli and X.\u00a0Cabr\u00e9. Fully nonlinear elliptic equations, volume\u00a043 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 1995.","DOI":"10.1090\/coll\/043"},{"key":"9347_CR10","doi-asserted-by":"crossref","unstructured":"L.\u00a0Caffarelli, M.G.\u00a0Crandall, M.\u00a0Kocan, and A.\u00a0\u015awiech, On viscosity solutions of fully nonlinear equations with measurable ingredients. Commun. Pure Appl. Math., 1996.","DOI":"10.1002\/(SICI)1097-0312(199604)49:4<365::AID-CPA3>3.3.CO;2-V"},{"key":"9347_CR11","doi-asserted-by":"crossref","unstructured":"L.\u00a0Caffarelli and L.\u00a0Silvestre. Smooth approximations of solutions to nonconvex fully nonlinear elliptic equations. In Nonlinear partial differential equations and related topics, vol. 229, pp. 67\u201385. Am. Math. Soc., Providence, RI, 2010.","DOI":"10.1090\/trans2\/229\/05"},{"key":"9347_CR12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1002\/cpa.20208","volume":"61","author":"L Caffarelli","year":"2008","unstructured":"L.\u00a0Caffarelli and P.\u00a0E. Souganidis. A rate of convergence for monotone finite difference approximations to fully nonlinear, uniformly elliptic PDEs. Comm. Pure Appl. Math., 61:1\u201317, 2008.","journal-title":"Commun. Pure Appl. Math."},{"issue":"1","key":"9347_CR13","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1051\/m2an\/1995290100971","volume":"29","author":"F Camilli","year":"1995","unstructured":"F.\u00a0Camilli and M.\u00a0Falcone. An approximation scheme for the optimal control of diffusion processes. RAIRO Mod\u00e9l. Math. Anal. Num\u00e9r., 29(1):97\u2013122, 1995.","journal-title":"RAIRO Mod\u00e9l. Math. Anal. Num\u00e9r."},{"issue":"1","key":"9347_CR14","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1016\/0045-7825(73)90019-4","volume":"2","author":"Ph Ciartet","year":"1973","unstructured":"Ph.\u00a0Ciartet and P.A.\u00a0Raviart. Maximum principle and uniform convergence for the finite element method. Comp. Meths. Appl. Mech. Eng., 2(1):17-31. 1973.","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"1","key":"9347_CR15","first-page":"149","volume":"40","author":"F Chiarenza","year":"1991","unstructured":"F.\u00a0Chiarenza, M.\u00a0Frasca, and P.\u00a0Longo. Interior $$W^2_p$$ W p 2 estimates for nondivergence elliptic equations with discontinuous coefficients. Ricerche Mat., 40(1):149\u2013168, 1991.","journal-title":"Ricerche Mat."},{"issue":"2","key":"9347_CR16","first-page":"841","volume":"336","author":"F Chiarenza","year":"1993","unstructured":"F.\u00a0Chiarenza, M.\u00a0Frasca, and P.\u00a0Longo. $$W^2_p$$ W p 2 -solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients. Trans. Amer. Math. Soc., 336(2):841\u2013853, 1993.","journal-title":"Trans. Am. Math. Soc."},{"key":"9347_CR17","doi-asserted-by":"crossref","unstructured":"P.\u00a0G. Ciarlet. Basic error estimates for elliptic problems. In Handbook of numerical analysis, Vol. II, Handb. Numer. Anal., II, pp. 17\u2013351. North-Holland, Amsterdam, 1991.","DOI":"10.1016\/S1570-8659(05)80039-0"},{"issue":"9","key":"9347_CR18","doi-asserted-by":"crossref","first-page":"779","DOI":"10.1016\/S1631-073X(03)00149-3","volume":"336","author":"EJ Dean","year":"2003","unstructured":"E.\u00a0J. Dean and R.\u00a0Glowinski. Numerical solution of the two-dimensional elliptic Monge-Amp\u00e8re equation with Dirichlet boundary conditions: an augmented Lagrangian approach. C. R. Math. Acad. Sci. Paris, 336(9):779\u2013784, 2003.","journal-title":"C. R. Math. Acad. Sci. Paris"},{"issue":"283","key":"9347_CR19","doi-asserted-by":"crossref","first-page":"1433","DOI":"10.1090\/S0025-5718-2012-02632-9","volume":"82","author":"K Debrabant","year":"2013","unstructured":"K.\u00a0Debrabant and E.\u00a0R. Jakobsen. Semi-Lagrangian schemes for linear and fully non-linear diffusion equations. Math. Comp., 82(283):1433\u20131462, 2013.","journal-title":"Math. Comput."},{"key":"9347_CR20","unstructured":"X.\u00a0Feng, L.\u00a0Hennings, and M.\u00a0Neilan. Finite element methods for second order linear elliptic partial differential equations in non-divergence form. Math. Comput., (to appear)."},{"key":"9347_CR21","unstructured":"W.\u00a0H. Fleming and H.\u00a0M. Soner. Controlled Markov processes and viscosity solutions, volume\u00a025 of Stochastic Modelling and Applied Probability. Springer, New York, Second Edition, 2006."},{"key":"9347_CR22","unstructured":"D.\u00a0Gilbarg and N.\u00a0S. Trudinger. Elliptic partial differential equations of second order, volume 224 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin, Second Edition, 1983."},{"key":"9347_CR23","doi-asserted-by":"crossref","unstructured":"B. Gr\u00fcnbaum. Convex polytopes, volume 221 of Graduate Texts in Mathematics. Springer, New York, Second Edition, 2003. Prepared and with a preface by Volker Kaibel, Victor Klee and G\u00fcnter M. Ziegler.","DOI":"10.1007\/978-1-4613-0019-9"},{"key":"9347_CR24","unstructured":"Q. Han and F. Lin. Elliptic partial differential equations, volume\u00a01 of Courant Lecture Notes in Mathematics. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1997."},{"issue":"3","key":"9347_CR25","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1007\/s00041-001-4031-6","volume":"2","author":"RR Jensen","year":"1995","unstructured":"R.\u00a0R. Jensen. Uniformly elliptic PDEs with bounded, measurable coefficients. J. Fourier Anal. Appl., 2(3):237\u2013259, 1995.","journal-title":"J. Fourier Anal. Appl."},{"issue":"1","key":"9347_CR26","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1137\/110856198","volume":"51","author":"M Jensen","year":"2013","unstructured":"M.\u00a0Jensen and I.\u00a0Smears. On the convergence of finite element methods for Hamilton-Jacobi-Bellman equations. SIAM J. Numer. Anal., 51(1):137\u2013162, 2013.","journal-title":"SIAM J. Numer. Anal."},{"key":"9347_CR27","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/s11118-006-9034-0","volume":"26","author":"D Kim","year":"2007","unstructured":"D. Kim. Second order elliptic equations in $${\\mathbb{R}}^d$$ R d with piecewise continuous coefficients. Potential Anal., 26:189\u2013212, 2007.","journal-title":"Potential Anal."},{"key":"9347_CR28","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1007\/s002110050160","volume":"72","author":"M Kocan","year":"1995","unstructured":"M. Kocan. Approximation of viscosity solutions of elliptic partial differential equations on minimal grids Numer. Math., 72:73\u201392, 1995.","journal-title":"Numer. Math."},{"key":"9347_CR29","doi-asserted-by":"crossref","unstructured":"S.\u00a0Korotov, M.\u00a0K\u0159\u00ed\u017eek, and P.\u00a0Neittaanm\u00e4ki. Weakened acute type condition for tetrahedral triangulations and the discrete maximum principle. Math. Comput., 70(233):107\u2013119 (electronic), 2001.","DOI":"10.1090\/S0025-5718-00-01270-9"},{"key":"9347_CR30","first-page":"245","volume":"9","author":"NV Krylov","year":"1997","unstructured":"N.\u00a0V. Krylov. On the rate of convergence of finite-difference approximations for Bellman\u2019s equations. Algebra i Analiz, 9:245\u2013256, 1997.","journal-title":"Algebra i Analiz"},{"key":"9347_CR31","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s004400050264","volume":"117","author":"NV Krylov","year":"2000","unstructured":"N.\u00a0V. Krylov. On the rate of convergence of finite-difference approximations for Bellman\u2019s equations with variable coefficients. Probab. Theory Related Fields, 117:1\u201316, 2000.","journal-title":"Probab. Theory Related Fields"},{"key":"9347_CR32","unstructured":"A.\u00a0Kufner, O.\u00a0John, and S.\u00a0Fu\u010d\u00edk. Function spaces. Noordhoff International Publishing, Leyden; Academia, Prague, 1977. Monographs and Textbooks on Mechanics of Solids and Fluids; Mechanics: Analysis."},{"key":"9347_CR33","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1137\/0729008","volume":"29","author":"HJ Kuo","year":"1992","unstructured":"H.\u00a0J. Kuo and N.\u00a0S. Trudinger. Discrete methods for fully nonlinear elliptic equations. SIAM J. Numer. Anal., 29:123\u2013135, 1992.","journal-title":"SIAM J. Numer. Anal."},{"key":"9347_CR34","doi-asserted-by":"crossref","unstructured":"H-J. Kuo and N.\u00a0S. Trudinger. A note on the discrete Aleksandrov-Bakelman maximum principle. In Proceedings of 1999 International Conference on Nonlinear Analysis (Taipei), vol.\u00a04, pp. 55\u201364, 2000.","DOI":"10.11650\/twjm\/1500407198"},{"key":"9347_CR35","doi-asserted-by":"crossref","unstructured":"H.\u00a0J. Kushner and P.\u00a0Dupuis. Numerical methods for stochastic control problems in continuous time, volume\u00a024 of Applications of Mathematics (New York). Springer, New York, Second Edition, 2001. Stochastic Modelling and Applied Probability.","DOI":"10.1007\/978-1-4613-0007-6"},{"key":"9347_CR36","unstructured":"O.\u00a0A. Ladyzhenskaya and N.\u00a0N. Ural\u2019tseva. Linear and quasilinear elliptic equations. Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis. Academic Press, New York-London, 1968."},{"issue":"2","key":"9347_CR37","doi-asserted-by":"crossref","first-page":"786","DOI":"10.1137\/100787672","volume":"33","author":"O Lakkis","year":"2011","unstructured":"O.\u00a0Lakkis and T.\u00a0Pryer. A finite element method for second order nonvariational elliptic problems. SIAM J. Sci. Comput., 33(2):786\u2013801, 2011.","journal-title":"SIAM J. Sci. Comput."},{"issue":"3","key":"9347_CR38","first-page":"839","volume":"26","author":"A Lorenzi","year":"1972","unstructured":"A. Lorenzi. On elliptic equations with piecewise constant coefficients, II. Ann. Scuola Norm. Sup. Pisa, 26(3), 839-870, 1972.","journal-title":"Ann. Scuola Norm. Sup. Pisa"},{"key":"9347_CR39","doi-asserted-by":"crossref","unstructured":"A.\u00a0Maugeri, D.\u00a0K. Palagachev, and L.\u00a0G. Softova. Elliptic and parabolic equations with discontinuous coefficients, volume 109 of Mathematical Research. Wiley-VCH Verlag Berlin GmbH, Berlin, 2000.","DOI":"10.1002\/3527600868"},{"key":"9347_CR40","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1002\/sapm1952311253","volume":"31","author":"TS Motzkin","year":"1953","unstructured":"T.\u00a0S. Motzkin and W. Wasow. On the approximation of linear elliptic differential equations by difference equations with positive coefficients. J. Math. Physics, 31:253\u201359, 1953.","journal-title":"J. Math. Phys."},{"issue":"3","key":"9347_CR41","first-page":"537","volume":"24","author":"N Nadirashvili","year":"1997","unstructured":"N.\u00a0Nadirashvili. Nonuniqueness in the martingale problem and the Dirichlet problem for uniformly elliptic operators. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24(3):537\u2013549, 1997.","journal-title":"Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)"},{"key":"9347_CR42","doi-asserted-by":"crossref","unstructured":"R.\u00a0H. Nochetto, M.\u00a0Paolini, and C.\u00a0Verdi. An adaptive finite element method for two-phase Stefan problems in two space dimensions. I. Stability and error estimates. Math. Comput., 57(195):73\u2013108, S1\u2013S11, 1991.","DOI":"10.1090\/S0025-5718-1991-1079028-X"},{"key":"9347_CR43","unstructured":"R.\u00a0H.\u00a0Nochetto and W.\u00a0Zhang. Pointwise rates of convergence for the Oliker-Prussner method for the Monge-Amp\u00e8re equation. (submitted)."},{"key":"9347_CR44","unstructured":"R.\u00a0H.\u00a0Nochetto and W.\u00a0Zhang. Two-scale FEM for equations in non-divergence form: Calder\u00f3n-Zygmund theory. (in preparation)."},{"key":"9347_CR45","doi-asserted-by":"crossref","unstructured":"M.\u00a0Safonov. Nonuniqueness for second-order elliptic equations with measurable coefficients. SIAM J. Math. Anal., 30(4):879\u2013895 (electronic), 1999.","DOI":"10.1137\/S0036141096309046"},{"issue":"157","key":"9347_CR46","first-page":"1","volume":"38","author":"AH Schatz","year":"1982","unstructured":"A.\u00a0H. Schatz and L.\u00a0B. Wahlbin. On the quasi-optimality in $$L_{\\infty }$$ L \u221e of the $$\\dot{H}^{1}$$ H \u02d9 1 -projection into finite element spaces. Math. Comp., 38(157):1\u201322, 1982.","journal-title":"Math. Comput."},{"issue":"4","key":"9347_CR47","doi-asserted-by":"crossref","first-page":"2088","DOI":"10.1137\/120899613","volume":"51","author":"I Smears","year":"2013","unstructured":"I.\u00a0Smears and E.\u00a0S\u00fcli. Discontinuous Galerkin finite element approximation of nondivergence form elliptic equations with Cord\u00e8s coefficients. SIAM J. Numer. Anal., 51(4):2088\u20132106, 2013.","journal-title":"SIAM J. Numer. Anal."},{"issue":"2","key":"9347_CR48","doi-asserted-by":"crossref","first-page":"993","DOI":"10.1137\/130909536","volume":"52","author":"I Smears","year":"2014","unstructured":"I.\u00a0Smears and E.\u00a0S\u00fcli. Discontinuous Galerkin finite element approximation of Hamilton-Jacobi-Bellman equations with Cord\u00e8s coefficients. SIAM J. Numer. Anal., 52(2):993\u20131016, 2014.","journal-title":"SIAM J. Numer. Anal."},{"key":"9347_CR49","unstructured":"A.\u00a0H. Stroud. Approximate calculation of multiple integrals. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. Prentice-Hall Series in Automatic Computation."},{"key":"9347_CR50","unstructured":"S. Walker. FELICITY: Finite Element Implementation and Computational Interface Tool for you, Tutorial, 2013."},{"key":"9347_CR51","unstructured":"C.\u00a0Wang and J.\u00a0Wang. A primal-dual weak galerkin finite element method for second order elliptic equations in non-divergence form. http:\/\/arxiv.org\/abs\/1510.03499 ."}],"container-title":["Foundations of Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10208-017-9347-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-017-9347-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-017-9347-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,10,3]],"date-time":"2020-10-03T07:47:30Z","timestamp":1601711250000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10208-017-9347-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,3,1]]},"references-count":51,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2018,6]]}},"alternative-id":["9347"],"URL":"https:\/\/doi.org\/10.1007\/s10208-017-9347-y","relation":{},"ISSN":["1615-3375","1615-3383"],"issn-type":[{"value":"1615-3375","type":"print"},{"value":"1615-3383","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,3,1]]}}}