2018/01/03 by R. Nochetto, Ricardo H. Nochetto, D. Ntogkas +3 · 39 citations
Mathematics · #Applied mathematics #Boundary (topology) #Boundary value problem #Convergence (economics) #Dimension (graph theory) #Dirichlet boundary condition #Dirichlet problem #Domain (mathematical analysis) #Finite element method #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical analysis #Mathematics #Nonlinear Partial Differential Equations #Piecewise linear function #Pure mathematics #Scale (ratio) #Stencil #Viscosity #Viscosity solution
paper · pdf · doi:10.1090/mcom/3353
published in Mathematics of Computation 88(316), 637-664 (American Mathematical Society)
openalex publication_date 2018/01/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
We propose a two-scale finite element method for the Monge-Ampère equation with Dirichlet boundary condition in dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d greater-than-or-equal-to 2"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">d≥ 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and prove that it converges to the viscosity solution uniformly. The method is inspired by a finite difference method of Froese and Oberman, but is defined on unstructured grids and relies on two separate scales: the first one is the mesh size <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h"> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding="application/x-tex">h</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the second one is a larger scale that controls appropriate directions and substitutes the need of a wide-stencil. The main tools for the analysis are a discrete comparison principle and discrete barrier functions that control the behavior of the discrete solution, which is continuous piecewise linear, both close to the boundary and in the interior of the domain.