Michael Neilan

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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, 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(D2u0) = 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)
This article surveys the recent developments in computational methods for second order fully nonlinear partial differential equations (PDEs), a relatively new subarea within numerical PDEs. Due to their ever increasing importance in mathematics itself (e.g., differential geometry and PDEs) and in many scientific and engineering fields (e.g., astrophysics,(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)
In this paper, we study finite element approximations of the viscosity solution of the fully nonlinear Monge-Ampère equation, det(Du) = f (> 0) using the well-known nonconforming Morley element. Our approach is based on the vanishing moment method, which was recently proposed as a constructive way to approximate fully nonlinear second order equations by the(More)
Two families of conforming finite elements for the two-dimensional Stokes problem are developed, guided by two discrete smoothed de Rham complexes, which we coin “Stokes complexes.” We show that the finite element pairs are inf-sup stable and also provide pointwise mass conservation on very general triangular meshes.
In this paper we develop two conforming finite element methods for a fourth order bi-wave equation arising as a simplified Ginzburg-Landau-type model for d-wave superconductors in absence of applied magnetic field. Unlike the biharmonic operator ∆2, the bi-wave operator 2 is not an elliptic operator, so the energy space for the bi-wave equation is much(More)
In this paper, we develop and analyze C0 penalty methods for the fully nonlinear Monge-Ampère equation det(D2u) = 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)