This paper studies mixed finite element approximations of the viscosity solution to the Dirichlet problem for the fully nonlinear Mongeâ€“AmpÃ¨re equation det(D2u0) = f (> 0) based on the vanishingâ€¦ (More)

This paper concerns with numerical approximations of solutions of fully nonlinear second order partial differential equations (PDEs). A new notion of weak solutions, called moment solutions, isâ€¦ (More)

This paper develops and analyzes some interior penalty discontinuous Galerkin methods using piecewise linear polynomials for the Helmholtz equation with the first order absorbing boundary conditionâ€¦ (More)

In this paper we present PDE and finite element analyses for a system of PDEs consisting of the Darcy equation and the Cahnâ€“Hilliard equation, which arises as a diffuse interface model for theâ€¦ (More)

This paper develops and analyzes finite element Galerkin and spectral Galerkin methods for approximating viscosity solutions of the fully nonlinear Monge-AmpÃ¨re equation det(D2u0) = f (> 0) based onâ€¦ (More)

In this first part of a series, we propose and analyze, under minimum regularity assumptions, a semi-discrete (in time) scheme and a fully discrete mixed finite element scheme for the Cahn-Hilliardâ€¦ (More)

This paper develops and analyzes two fully discrete interior penalty discontinuous Galerkin (IP-DG) methods for the Allen-Cahn equation, which is a nonlinear singular perturbation of the heatâ€¦ (More)