On Finite Element Methods for Fully Nonlinear Elliptic Equations of Second Order

Böhmer, Klaus

doi:10.1137/040621740

Cited by 67 publications

(70 citation statements)

References 27 publications

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“…Böhmer [3] performs consistency and stability analysis of finite element methods for fully nonlinear equations. However, as mentioned before, consistency and stability is not enough to prove convergence to viscosity solutions.…”

Section: Related Work: Other Methodsmentioning

confidence: 99%

Two Numerical Methods for the elliptic Monge-Ampère equation

Benamou¹,

Froese²,

Oberman³

2010

ESAIM: M2AN

105

148

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Abstract. The numerical solution of the elliptic Monge-Ampère Partial Differential Equation hasbeen a subject of increasing interest recently [Glowinski, in 6th International Congress on Industrial and Applied Mathematics, ICIAM 07, Invited Lectures (2009) 155-192; Oliker and Prussner, Numer. Math. 54 (1988) which converge even for singular solutions. However, many of the newly proposed methods lack numerical evidence of convergence on singular solutions, or are known to break down in this case. In this article we present and study the performance of two methods. The first method, which is simply the natural finite difference discretization of the equation, is demonstrated to be the best performing method (in terms of convergence and solution time) currently available for generic (possibly singular) problems, in particular when the right hand side touches zero. The second method, which involves the iterative solution of a Poisson equation involving the Hessian of the solution, is demonstrated to be the best performing (in terms of solution time) when the solution is regular, which occurs when the right hand side is strictly positive.

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Section: Related Work: Other Methodsmentioning

confidence: 99%

Two Numerical Methods for the elliptic Monge-Ampère equation

Benamou¹,

Froese²,

Oberman³

2010

ESAIM: M2AN

105

148

View full text Add to dashboard Cite

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“…In order to avoid the complications associated with the construction of finite element sub-spaces of H 2 (Ω) (see, however, [5,25] for such an approach), we employ here a mixed finite element approximation (closely related to those discussed in, e.g., [22,23,31,36,47] for the solution of linear and nonlinear bi-harmonic problems). Following this approach, it is possible to solve (2.1) employing approximations commonly used for the solution of second order elliptic problems (piecewise linear and globally continuous over a triangulation of Ω for example).…”

Section: Generalitiesmentioning

confidence: 99%

“…As such, it has recently received considerable attention from both the analytical and computational standpoints as shown by, e.g., [2,5,6,14,24,37,40,41,43,44,50], with applications in geometry, mechanics and physics.…”

Section: Introductionmentioning

confidence: 99%

A least-squares method for the numerical solution of the Dirichlet problem for the elliptic monge − ampère equation in dimension two

Caboussat¹,

Glowinski²,

Sorensen³

2013

ESAIM: COCV

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Abstract. We address in this article the computation of the convex solutions of the Dirichlet problem for the real elliptic Monge−Ampère equation for general convex domains in two dimensions. The method we discuss combines a least-squares formulation with a relaxation method. This approach leads to a sequence of Poisson−Dirichlet problems and another sequence of low dimensional algebraic eigenvalue problems of a new type. Mixed finite element approximations with a smoothing procedure are used for the computer implementation of our least-squares/relaxation methodology. Domains with curved boundaries are easily accommodated. Numerical experiments show the convergence of the computed solutions to their continuous counterparts when such solutions exist. On the other hand, when classical solutions do not exist, our methodology produces solutions in a least-squares sense.Résumé. Nousétudions, dans cet article, une méthode numérique, pour le calcul des solutions convexes du problème de Dirichlet pour l'équation de Monge−Ampère elliptique, dans des domaines bi-dimensionnel convexes. Une méthode de moindres carrés est coupléeà un algorithme de relaxation, conduisantà la résolution d'une suite de problèmes de Poisson−Dirichlet, et d'une suite de problèmes de valeurs propres de petite dimension d'un type nouveau. Une approximation paréléments finis mixtes, coupléeà une méthode de régularisation, est utilisée pour implémenter la méthode de moindres-carrés/relaxation ci-dessus, de sorte que les domaines avec frontière courbe sont traités facilement. Des expériences numériques montrent la convergence des solutions calculées vers la solution convexe du problème continu, lorsqu'une telle solution existe. Par ailleurs, si le problème n'a pas de solution classique, notre méthodologie fournit des solutions au sens des moindres carrés.Mathematics Subject Classification. 65N30, 65K10, 65F30, 49M15, 49K20.

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“…The nonlinear problem (4)- (5) can be solved iteratively by a Newton method [1], where the initial guess u h 0 ∈ S h satisfies the boundary condition…”

Section: Theorem 1 ([1 Theorem 87] and [2 Theorem 52])mentioning

confidence: 99%

“…Recently, Böhmer [1,2] introduced a general approach that solves the Dirichlet problem for fully nonlinear elliptic equations numerically with the help of a sequence of linear elliptic equations used within an appropriate Newton scheme. These linear elliptic equations can be solved by the finite element method, but the discretisation has to be done by appropriate spaces of C 1 finite elements (splines) that admit a stable splitting into a subspace satisfying zero boundary conditions, and its complement.…”

Section: Introductionmentioning

confidence: 99%