Quadratic finite element approximations of the Monge-Ampère equation
type: Item
instance of: scholarly article
title: Quadratic finite element approximations of the Monge-Ampère equation
zbMATH Open document ID: 1272.65094
author: Michael Neilan
published in: Journal of Scientific Computing
publication date: 2013-02-04T00:00:00Z
review text: The paper deals with the Dirichlet boundary value problem for the fully nonlinear Monge-Ampère equation \[ \text{det}\big(D^2u\big)=f \; \text{in} \; \Omega, \] \[ u=g \; \text{on} \; \partial \Omega, \] where \(\Omega \in \mathbb{R}^2\) is a smooth strictly convex domain, \(f>0\), \(g\) are given functions, and \(\text{det}\big(D^2u\big)\) denotes the determinant of the Hessian matrix \(D^2u\) of the unknown function. The author considers the finite element method for this problem, shows the convergence of quadratic finite element approximations, obtains quasi-optimal error estimates in a \(W^{2,p}\)-type norm, localizes pointwise error estimates, and the convergence of Newton's method. Numerical experiments are presented which confirm the theoretical results.
reviewed by: Iwan Gawriljuk
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 35J96
Mathematics Subject Classification ID: 65N12
Mathematics Subject Classification ID: 65N15
zbMATH DE Number: 6134443
zbMATH Keywords: fully nonlinear Monge-Ampère equation
zbMATH Keywords: finite element method
zbMATH Keywords: convergence
zbMATH Keywords: quasi-optimal error estimates
zbMATH Keywords: localized pointwise error estimates
zbMATH Keywords: Dirichlet boundary value problem
zbMATH Keywords: Newton's method
zbMATH Keywords: numerical experiments
describes a project that uses: COMSOL
MaRDI profile type: Publication
full work available at URL: s10915-012-9617-4
OpenAlex ID: W2009553465
cites work: Q4073917
cites work: Optimal Finite-Element Interpolation on Curved Domains
cites work: Numerical solutions of boundary value problems for Κ-surfaces inR3
cites work: Weak Existence for the Semigeostrophic Equations Formulated as a Coupled Monge--Ampère/Transport Problem
cites work: On Finite Element Methods for Fully Nonlinear Elliptic Equations of Second Order
cites work: The Construction of Preconditioners for Elliptic Problems by Substructuring. I
cites work: The Mathematical Theory of Finite Element Methods
cites work: Poincaré--Friedrichs Inequalities for Piecewise H1 Functions
cites work: 𝒞⁰ penalty methods for the fully nonlinear Monge-Ampère equation
cites work: Finite element approximations of the three dimensional Monge-Ampère equation
cites work: The dirichlet problem for nonlinear second-order elliptic equations I. Monge-ampégre equation
cites work: Monge Ampère Equation: Applications to Geometry and Optimization
cites work: Pointwise Error Estimates of Discontinuous Galerkin Methods with Penalty for Second-Order Elliptic Problems
cites work: Q4162607
cites work: Q3999241
cites work: Numerical solution of the two-dimensional elliptic Monge--Ampère equation with Dirichlet boundary conditions: An augmented Lagrangian approach
cites work: Numerical solution of the two-dimensional elliptic Monge--Ampère equation with Dirichlet boundary conditions: a least-squares approach
cites work: An augmented Lagrangian approach to the numerical solution of the Dirichlet problem for the elliptic Monge-Ampère equation in two dimensions
cites work: Numerical methods for fully nonlinear elliptic equations of the Monge-Ampère type
cites work: An optimal robust equidistribution method for two-dimensional grid adaptation based on Monge-Kantorovich optimization
cites work: Sharply localized pointwise and $W_\infty^{-1}$ estimates for finite element methods for quasilinear problems
cites work: Q4952326
cites work: Vanishing moment method and moment solutions for fully nonlinear second order partial differential equations
cites work: Mixed Finite Element Methods for the Fully Nonlinear Monge–Ampère Equation Based on the Vanishing Moment Method
cites work: Analysis of Galerkin methods for the fully nonlinear Monge-Ampère equation
cites work: Convergent Finite Difference Solvers for Viscosity Solutions of the Elliptic Monge–Ampère Equation in Dimensions Two and Higher
cites work: Fast finite difference solvers for singular solutions of the elliptic Monge-Ampère equation
cites work: Elliptic partial differential equations of second order
cites work: Boundary-value problems on \(S^ n\) for surfaces of constant Gauss curvature
cites work: The Dirichlet problem for Monge-Ampère equations in non-convex domains and spacelike hypersurfaces of constant Gauss curvature
cites work: The existence of hypersurfaces of constant Gauss curvature with prescribed boundary.
cites work: The Monge-Ampère equation
cites work: Localized pointwise error estimates and global \(L^p\) error estimates for Nitsche's method
cites work: Q2550279
cites work: Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian
cites work: On the numerical solution of the equation \(\frac{\partial ^ 2z\partial ^ 2z}{\partial x^ 2\partial y^ 2}-(\frac{\partial ^ 2z}{\partial x\partial y})^ 2=f\) and its discretizations. I
cites work: Interior Maximum-Norm Estimates for Finite Element Methods, Part II
cites work: Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids: Part I. Global estimates
cites work: Q3536061
cites work: Q4805362
DOI: 10.1007/S10915-012-9617-4
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description: scientific article; zbMATH DE number 6134443