메뉴 건너뛰기




Volumn 46, Issue 3, 2008, Pages 1212-1249

On finite element methods for fully nonlinear elliptic equations of second order

Author keywords

C1 finite elements; Conforming; Convergence; Fully nonlinear elliptic differential equations; Nonconforming finite element method; Order 2; Quasi linear elliptic equations not in divergence form; Rn; Stability; Variational crimes

Indexed keywords

BOUNDARY VALUE PROBLEMS; COMPUTATIONAL MECHANICS; CONVERGENCE OF NUMERICAL METHODS; DIFFERENTIAL EQUATIONS; LINEAR EQUATIONS; LINEARIZATION; MATHEMATICAL OPERATORS; NEWTON-RAPHSON METHOD; NONLINEAR EQUATIONS; SYSTEM STABILITY;

EID: 55349084784     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/040621740     Document Type: Article
Times cited : (70)

References (64)
  • 1
    • 51249176387 scopus 로고
    • Regularity results for discrete solutions of second order elliptic problems in the finite element method
    • N. E. AGUILERA AND L. A. CAFFARELLI, Regularity results for discrete solutions of second order elliptic problems in the finite element method, Calcolo, 23 (1986), pp. 327-353.
    • (1986) Calcolo , vol.23 , pp. 327-353
    • AGUILERA, N.E.1    CAFFARELLI, L.A.2
  • 2
    • 55349126996 scopus 로고
    • A Mesh Independence Principle for Operator Equations and Their Discretization
    • Technical report, Department of Mathematics, Colorado State University, Fort Collins, CO
    • E. L. ALLGOWER AND K. BÖHMER, A Mesh Independence Principle for Operator Equations and Their Discretization, Technical report, Department of Mathematics, Colorado State University, Fort Collins, CO, 1984.
    • (1984)
    • ALLGOWER, E.L.1    BÖHMER, K.2
  • 3
    • 0023577191 scopus 로고
    • Application of the mesh independence principle to mesh refinement strategies
    • E. L. ALLGOWER AND K. BÖHMER, Application of the mesh independence principle to mesh refinement strategies, SIAM J. Numer. Anal., 24 (1987), pp. 1335-1351.
    • (1987) SIAM J. Numer. Anal , vol.24 , pp. 1335-1351
    • ALLGOWER, E.L.1    BÖHMER, K.2
  • 4
    • 0022661397 scopus 로고
    • A mesh-independence principle for operator equations and their discretization
    • E. L. ALLGOWER, K. BÖHMER, F. A. POTRA, AND W. C. RHEINBOLDT, A mesh-independence principle for operator equations and their discretization, SIAM J. Numer. Anal., 23 (1986), pp. 160-169.
    • (1986) SIAM J. Numer. Anal , vol.23 , pp. 160-169
    • ALLGOWER, E.L.1    BÖHMER, K.2    POTRA, F.A.3    RHEINBOLDT, W.C.4
  • 7
    • 0039705727 scopus 로고
    • On a numerical Liapunov-Schmidt method for operator equations
    • K. BÖHMER, On a numerical Liapunov-Schmidt method for operator equations, Computing, 51 (1993), pp. 237-269.
    • (1993) Computing , vol.51 , pp. 237-269
    • BÖHMER, K.1
  • 8
    • 55349121448 scopus 로고    scopus 로고
    • On numerical bifurcation studies for general operator equations
    • J. Sprekels, B. Fiedler, and K. Gröger, eds, World Scientific, Singapore
    • K. BÖHMER, On numerical bifurcation studies for general operator equations, in International Conference on Differential Equations, Vol. 2, J. Sprekels, B. Fiedler, and K. Gröger, eds., 1999, World Scientific, Singapore, 2000, pp. 877-883.
    • (1999) International Conference on Differential Equations , vol.2 , pp. 877-883
    • BÖHMER, K.1
  • 9
    • 4243380931 scopus 로고    scopus 로고
    • On hybrid methods for bifurcation studies for general operator equations
    • Systems, B. Fiedler, ed, Springer-Verlag, Berlin, Heidelberg, New York
    • K. BÖHMER, On hybrid methods for bifurcation studies for general operator equations, in Ergodic theory, Analysis, and Efficient Simulation of Dynamical Systems, B. Fiedler, ed., Springer-Verlag, Berlin, Heidelberg, New York, 2001, pp. 73-107.
    • (2001) Ergodic theory, Analysis, and Efficient Simulation of Dynamical , pp. 73-107
    • BÖHMER, K.1
  • 12
    • 0032677639 scopus 로고    scopus 로고
    • Numerical Liapunov-Schmidt spectral method for k-determined problems
    • K. BÖHMER AND N. SASSMANNSHAUSEN, Numerical Liapunov-Schmidt spectral method for k-determined problems, Comput. Methods Appl. Mech. Engrg., 170 (1999), pp. 277-312.
    • (1999) Comput. Methods Appl. Mech. Engrg , vol.170 , pp. 277-312
    • BÖHMER, K.1    SASSMANNSHAUSEN, N.2
  • 18
    • 55349131269 scopus 로고    scopus 로고
    • Y. Z. CHEN AND L. C. WU, Second Order Elliptic Equations and Elliptic Systems, Transl. Math. Monogr. 174, AMS, Providence, RI, 1998.
    • Y. Z. CHEN AND L. C. WU, Second Order Elliptic Equations and Elliptic Systems, Transl. Math. Monogr. 174, AMS, Providence, RI, 1998.
  • 20
    • 0016541865 scopus 로고
    • Approximation by finite element functions using local regularization
    • P. CLÉMENT, Approximation by finite element functions using local regularization, RAIRO Anal. Numer., 9 (1975), pp. 77-84.
    • (1975) RAIRO Anal. Numer , vol.9 , pp. 77-84
    • CLÉMENT, P.1
  • 21
    • 85095811894 scopus 로고    scopus 로고
    • The numerical analysis of bifurcation problems with application to fluid mechanics
    • Cambridge University Press, Cambridge, UK
    • K. A. CLIFF, A. SPENCE, AND S. J. TAVENER, The numerical analysis of bifurcation problems with application to fluid mechanics, in Acta Numerica, Cambridge University Press, Cambridge, UK, 2000, pp. 39-131.
    • (2000) Acta Numerica , pp. 39-131
    • CLIFF, K.A.1    SPENCE, A.2    TAVENER, S.J.3
  • 22
    • 0003346454 scopus 로고
    • Methods of Mathematical Physics
    • John Wiley & Sons, New York
    • R. COURANT AND D. HILBERT, Methods of Mathematical Physics. Volume II: Partial Differential Equations, John Wiley & Sons, New York, 1989.
    • (1989) Partial Differential Equations , vol.2
    • COURANT, R.1    HILBERT, D.2
  • 23
    • 46549092471 scopus 로고
    • Hamilton-Jacobi equations in infinite dimensions. 1: Uniqueness of viscosity solutions
    • M. G. CRANDALL AND P.-L. LIONS, Hamilton-Jacobi equations in infinite dimensions. 1: Uniqueness of viscosity solutions, J. Funct. Anal., 62 (1985), pp. 379-396.
    • (1985) J. Funct. Anal , vol.62 , pp. 379-396
    • CRANDALL, M.G.1    LIONS, P.-L.2
  • 24
    • 0030507383 scopus 로고    scopus 로고
    • Convergent difference schemes for nonlinear parabolic equations and mean curvature motion
    • M. G. CRANDALL AND P.-L. LIONS, Convergent difference schemes for nonlinear parabolic equations and mean curvature motion, Numer. Math., 75 (1996), pp. 17-41.
    • (1996) Numer. Math , vol.75 , pp. 17-41
    • CRANDALL, M.G.1    LIONS, P.-L.2
  • 25
    • 0005625355 scopus 로고    scopus 로고
    • M. G. CRANDALL, M. KOCAN, P. L. LIONS, AND A. SWIECH, Existence results for boundary problems for uniformly elliptic and parabolic fully nonlinear equations, Electron. J. Differential Equations, 1999.
    • M. G. CRANDALL, M. KOCAN, P. L. LIONS, AND A. SWIECH, Existence results for boundary problems for uniformly elliptic and parabolic fully nonlinear equations, Electron. J. Differential Equations, 1999.
  • 26
    • 55349099432 scopus 로고    scopus 로고
    • M. CROUZIEX AND P. A. RAVIART, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I., Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge, 3 (1973), pp. 33-75.
    • M. CROUZIEX AND P. A. RAVIART, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I., Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge, 3 (1973), pp. 33-75.
  • 27
    • 55349112474 scopus 로고    scopus 로고
    • Bericht Fachbereich Mathematik und Informatik 3, Philipps Universität Marburg, Marburg, Germany
    • O. DAVYDOV, Smooth Finite Elements and Stable Splitting, Bericht Fachbereich Mathematik und Informatik 3, Philipps Universität Marburg, Marburg, Germany, 2007.
    • (2007) Smooth Finite Elements and Stable Splitting
    • DAVYDOV, O.1
  • 28
    • 0345978401 scopus 로고    scopus 로고
    • Stable local bases for multivariate spline spaces
    • O. DAVYDOV, Stable local bases for multivariate spline spaces, J. Approx. Theory, 111 (2001), pp. 267-297.
    • (2001) J. Approx. Theory , vol.111 , pp. 267-297
    • DAVYDOV, O.1
  • 29
    • 0036114081 scopus 로고    scopus 로고
    • On stable local bases for bivariate polynomial spline spaces
    • O. DAVYDOV AND L. L. SCHUMAKER, On stable local bases for bivariate polynomial spline spaces, Constr. Approx., 18 (2002), pp. 87-116.
    • (2002) Constr. Approx , vol.18 , pp. 87-116
    • DAVYDOV, O.1    SCHUMAKER, L.L.2
  • 31
  • 32
    • 0038783139 scopus 로고    scopus 로고
    • Numerical solution of the two-dimensional elliptic Mange-Ampere equation with Dirichlet boundary conditions: An augmented Lagrangion approach
    • E. J. DEAN AND R. GLOWINSKI, Numerical solution of the two-dimensional elliptic Mange-Ampere equation with Dirichlet boundary conditions: An augmented Lagrangion approach, C. R. Math. Acad. Sci. Paris, 336 (2003), pp. 779-784.
    • (2003) C. R. Math. Acad. Sci. Paris , vol.336 , pp. 779-784
    • DEAN, E.J.1    GLOWINSKI, R.2
  • 33
    • 33646378906 scopus 로고    scopus 로고
    • An augmented Lagrangian approach to the numerical solution of the Dirichlet problem for the elliptic Monqe-Ampère equation in two dimensions
    • E. J. DEAN AND R. GLOWINSKI, An augmented Lagrangian approach to the numerical solution of the Dirichlet problem for the elliptic Monqe-Ampère equation in two dimensions, Electron. Trans. Numer. Anal., 22 (2006), pp. 71-96.
    • (2006) Electron. Trans. Numer. Anal , vol.22 , pp. 71-96
    • DEAN, E.J.1    GLOWINSKI, R.2
  • 34
    • 0027069897 scopus 로고
    • Multigrid solution of non-linear balance equations in meteorology
    • S. R. FULTON, Multigrid solution of non-linear balance equations in meteorology, Commun. Appl. Numer. Methods, 8 (1992), pp. 695-706.
    • (1992) Commun. Appl. Numer. Methods , vol.8 , pp. 695-706
    • FULTON, S.R.1
  • 36
    • 0034371147 scopus 로고    scopus 로고
    • Hybrid Galerkin boundary elements: Theory and implementation
    • I. G. GRAHAM, W. HACKBUSCH, AND S. A. SAUTER, Hybrid Galerkin boundary elements: Theory and implementation, Numer. Math, 86 (2000), pp. 139-172.
    • (2000) Numer. Math , vol.86 , pp. 139-172
    • GRAHAM, I.G.1    HACKBUSCH, W.2    SAUTER, S.A.3
  • 40
    • 84966247178 scopus 로고
    • Approximation methods for nonlinear problems with application to two-point boundary value problems
    • H. B. KELLER, Approximation methods for nonlinear problems with application to two-point boundary value problems, Math. Comp., 29 (1975), pp. 464-474.
    • (1975) Math. Comp , vol.29 , pp. 464-474
    • KELLER, H.B.1
  • 41
    • 0013024444 scopus 로고
    • Linear elliptic difference inequalities with random coefficients
    • H.-J. KUO AND N. S. TRUDINGER, Linear elliptic difference inequalities with random coefficients, Math. Comp., 55 (1990), pp. 37-53.
    • (1990) Math. Comp , vol.55 , pp. 37-53
    • KUO, H.-J.1    TRUDINGER, N.S.2
  • 42
    • 0022734898 scopus 로고
    • Optimal isoparametric finite elements and error estimates for domains involvinq curved boundaries
    • M. LENOIR, Optimal isoparametric finite elements and error estimates for domains involvinq curved boundaries, SIAM J. Numer. Anal., 23 (1986), pp. 562-580.
    • (1986) SIAM J. Numer. Anal , vol.23 , pp. 562-580
    • LENOIR, M.1
  • 44
    • 34249964909 scopus 로고    scopus 로고
    • 2 = f and its discretizations. I., Numer. Math., 54 (1988), pp. 271-293.
    • 2 = f and its discretizations. I., Numer. Math., 54 (1988), pp. 271-293.
  • 48
    • 84966200902 scopus 로고
    • Finite element interpolation of nonsmooth functions satisfying boundary conditions
    • L. R. SCOTT AND S. ZHANG, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp., 54 (1990), pp. 483-493.
    • (1990) Math. Comp , vol.54 , pp. 483-493
    • SCOTT, L.R.1    ZHANG, S.2
  • 52
    • 30244448497 scopus 로고
    • Diskrete Konvergenz linearer Operatoren I
    • F. STUMMEL, Diskrete Konvergenz linearer Operatoren I, Math. Ann., 190 (1970), pp. 45-92.
    • (1970) Math. Ann , vol.190 , pp. 45-92
    • STUMMEL, F.1
  • 53
    • 30244520551 scopus 로고
    • Diskrete Konvergenz linearer Operatoren II
    • F. STUMMEL, Diskrete Konvergenz linearer Operatoren II, Math. Z., 120 (1971), pp. 231-264.
    • (1971) Math. Z , vol.120 , pp. 231-264
    • STUMMEL, F.1
  • 54
    • 0003278083 scopus 로고    scopus 로고
    • Partial Differential Equations I
    • Springer-Verlag, New York, Berlin, Heidelberg
    • M. E. TAYLOR, Partial Differential Equations I, Basic Theory, Springer-Verlag, New York, Berlin, Heidelberg, 1996.
    • (1996) Basic Theory
    • TAYLOR, M.E.1
  • 55
    • 55349101574 scopus 로고    scopus 로고
    • M. E. TAYLOR, Partial Differential Equations 11, Qualitative Studies of Linear Equations, Springer-Verlag, New York, Berlin, Heidelberg, 1996.
    • M. E. TAYLOR, Partial Differential Equations 11, Qualitative Studies of Linear Equations, Springer-Verlag, New York, Berlin, Heidelberg, 1996.
  • 56
    • 0003419901 scopus 로고    scopus 로고
    • Partial Differential Equations III
    • Springer-Verlag, New York, Berlin, Heidelberg
    • M. E. TAYLOR, Partial Differential Equations III, Nonlinear Equations, Springer-Verlag, New York, Berlin, Heidelberg, 1996.
    • (1996) Nonlinear Equations
    • TAYLOR, M.E.1
  • 57
    • 0035624933 scopus 로고    scopus 로고
    • Shape calculus and finite element method in smooth domains
    • T. TIIHONEN, Shape calculus and finite element method in smooth domains, Math. Comp., 70 (2001), pp. 1-15.
    • (2001) Math. Comp , vol.70 , pp. 1-15
    • TIIHONEN, T.1
  • 61
    • 0000802762 scopus 로고
    • An elliptic collocation-finite element method with interior penalties
    • M. F. WHEELER, An elliptic collocation-finite element method with interior penalties, SIAM J. Numer. Anal., 15 (1978), pp. 152-161.
    • (1978) SIAM J. Numer. Anal , vol.15 , pp. 152-161
    • WHEELER, M.F.1
  • 62
    • 0003297920 scopus 로고
    • Nonlinear Functional Analysis and Its Applications I
    • Springer-Verlag, New York, Berlin, Heidelberg, London, Paris, Tokyo
    • E. ZEIDLER, Nonlinear Functional Analysis and Its Applications I, Fixed-Point Theorems, Springer-Verlag, New York, Berlin, Heidelberg, London, Paris, Tokyo, 1986, 1990.
    • (1986) Fixed-Point Theorems
    • ZEIDLER, E.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.