212 citations · 505 across the 22 of their papers we have counts for
26 papers
On uniqueness of weak solutions to the second boundary value problem for generated prescribed Jacobian equations
Gerard Awanou, Gantumur Tsogtgerel
We prove that two Aleksandrov solutions of a generated prescribed Jacobian equation have the same gradients at points where they are both differentiable. For the optimal transporta…
Hybridization and postprocessing in finite element exterior calculus
Gerard Awanou, Maurice Fabien, Johnny Guzmán +1
We hybridize the methods of finite element exterior calculus for the Hodge-Laplace problem on differential -forms in . In the cases and , we recover…
Convergence of a damped Newton's method for discrete Monge-Ampere functions with a prescribed asymptotic cone
Gerard Awanou
For finite difference discretizations with linear complexity and provably convergent to weak solutions of the second boundary value problem for the Monge-Ampère equation, we give t…
The second boundary value problem for a discrete Monge-Ampere equation
Gerard Awanou
In this work we propose a discretization of the second boundary condition for the Monge-Ampere equation arising in geometric optics and optimal transport. The discretization we pro…
On weak convergence of Monge-Ampere measures for discrete convex mesh functions
Gerard Awanou
To a mesh function we associate the natural analogue of the Monge-Ampere measure. The latter is shown to be equivalent to the Monge-Ampere measure of the convex envelope. We prove…
Uniform limit of discrete convex functions
Gerard Awanou
We consider mesh functions which are discrete convex in the sense that their central second order directional derivatives are positive. Analogous to the case of a uniformly bounded…
A two-grid method for the interior penalty discretization of the Monge-Ampère equation
Gerard Awanou, Hengguang Li, Eric Malitz
The purpose of this paper is to analyze an efficient method for the solution of the nonlinear system resulting from the discretization of the elliptic Monge-Ampère equation by a $C…
A variational method for computing numerical solutions of the Monge-Ampere equation
Gerard Awanou, Leopold Matamba Messi
We present a numerical method for solving the Monge-Ampere equation based on the characterization of the solution of the Dirichlet problem as the minimizer of a convex functional o…
Convergence of finite difference schemes to the Aleksandrov solution of the Monge-Ampere equation
Gerard Awanou, Romeo Awi
We present a technique for proving convergence to the Aleksandrov solution of the Monge-Ampere equation of a stable and consistent finite difference scheme. We also require a notio…
Discrete Aleksandrov solutions of the Monge-Ampere equation
Gerard Awanou
We prove the convergence of a wide stencil finite difference scheme to the Aleksandrov solution of the elliptic Monge-Ampere equation when the right hand side is a sum of Dirac mas…
Standard finite elements for the numerical resolution of the elliptic Monge-Ampere equation: mixed methods
Gerard Awanou
We prove a convergence result for a mixed finite element method for the Monge-Ampere equation to its weak solution in the sense of Aleksandrov. The unknowns in the formulation are…
Iterative methods for k-Hessian equations
Gerard Awanou
On a domain of the n-dimensional Euclidean space, and for an integer k=1,...,n, the k-Hessian equations are fully nonlinear elliptic equations for k >1 and consist of the Poisson e…
Quadratic Mixed finite element approximations of the Monge-Ampere equation in 2D
Gerard Awanou
We give error estimates for a mixed finite element approximation of the two-dimensional elliptic Monge-Ampere equation with the unknowns approximated by Lagrange finite elements of…
Convergence of a hybrid scheme for the elliptic Monge-Ampere equation
Gerard Awanou
We prove the convergence of a hybrid discretization to the viscosity solution of the elliptic Monge-Ampere equation. The hybrid discretization uses a standard finite difference dis…
On standard finite difference discretizations of the elliptic Monge-Ampere equation
Gerard Awanou
Given an orthogonal lattice with mesh length h on a bounded convex domain, we propose to approximate the Aleksandrov solution of the Monge-Ampere equation by regularizing the data…
Standard finite elements for the numerical resolution of the elliptic Monge-Ampere equation: classical solutions
Gerard Awanou
We propose a new variational formulation of the elliptic Monge-Ampere equation and show how classical Lagrange elements can be used for the numerical resolution of classical soluti…
Standard finite elements for the numerical resolution of the elliptic Monge-Ampere equation: Aleksandrov solutions
Gerard Awanou
We prove a convergence result for a natural discretization of the Dirichlet problem of the elliptic Monge-Ampere equation using finite dimensional spaces of piecewise polynomial C0…
Mixed finite elements for elasticity on quadrilateral meshes
Douglas N. Arnold, Gerard Awanou, Weifeng Qiu
We present stable mixed finite elements for planar linear elasticity on general quadrilateral meshes. The symmetry of the stress tensor is imposed weakly and so there are three pri…
Pseudo transient continuation and time marching methods for Monge-Ampere type equations
Gerard Awanou
We present two numerical methods for the fully nonlinear elliptic Monge-Ampere equation. The first is a pseudo transient continuation method and the second is a pure pseudo time ma…
Nonconforming tetrahedral mixed finite elements for elasticity
Douglas N. Arnold, Gerard Awanou, Ragnar Winther
This paper presents a nonconforming finite element approximation of the space of symmetric tensors with square integrable divergence, on tetrahedral meshes. Used for stress approxi…
Finite element differential forms on cubical meshes
Douglas N. Arnold, Gerard Awanou
We develop a family of finite element spaces of differential forms defined on cubical meshes in any number of dimensions. The family contains elements of all polynomial degrees and…
The serendipity family of finite elements
Douglas N. Arnold, Gerard Awanou
We give a new, simple, dimension-independent definition of the serendipity finite element family. The shape functions are the span of all monomials which are linear in at least s-r…
Rectangular Mixed Elements for Elasticity with Weakly Imposed symmetry Condition
Gerard Awanou
We present new rectangular mixed finite elements for linear elasticity. The approach is based on a modification of the Hellinger-Reissner functional in which the symmetry of the st…
Spline element method for the Monge-Ampere equation
Gerard Awanou
We analyze the convergence of an iterative method for solving the nonlinear system resulting from a natural discretization of the Monge-Ampère equation with conforming approx…