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This paper concerns with numerical approximations of solutions of fully non-linear second order partial differential equations (PDEs). A new notion of weak solutions, called moment solutions, is introduced for fully nonlinear second order PDEs. Unlike viscosity solutions, moment solutions are defined by a constructive method, called the vanishing moment(More)
This paper develops and analyzes finite element Galerkin and spectral Galerkin methods for approximating viscosity solutions of the fully nonlinear Monge-Ampère equation det(D 2 u 0) = f (> 0) based on the vanishing moment method which was developed by the authors in [17, 15]. In this approach, the Monge-Ampère equation is approximated by the fourth order(More)
In this paper, we develop and analyze C 0 penalty methods for the fully nonlinear Monge-Ampère equation det(D 2 u) = f in two dimensions. The key idea in designing our methods is to build discretizations such that the resulting discrete linearizations are symmetric, stable, and consistent with the continuous linearization. We are then able to show the(More)
This paper develops and analyzes a finite element method for a nonlinear singular elliptic equation arising from the black hole theory in the general relativity. The nonlinear equation, which was derived and analyzed by Huisken and Ilmanen in [14], represents a level set formulation for the inverse mean curvature flow describing the evolution of a(More)
We present a family of conforming finite elements for the Stokes problem on general triangular meshes in two dimensions. The lowest order case consists of enriched piecewise linear polynomials for the velocity and piecewise constant polynomials for the pressure. We show that the elements satisfy the inf-sup condition and converges optimally for both the(More)