Wujun Zhang

Ricardo H. Nochetto, Dimitrios Ntogkas, Wujun Zhang

We propose a two-scale finite element method for the Monge-Ampère equation with Dirichlet boundary condition in dimension $d\ge2$ and prove that it converges to the viscosity solution uniformly. The method is inspired by a finite difference method of Froese and Oberman, but is defined on unstructured grids and relies on two separate scales: the first one is the mesh size $h$ and the second one is a larger scale that controls appropriate directions and substitutes the need of a wide stencil. The main tools for the analysis are a discrete comparison principle and discrete barrier functions that control the behavior of the discrete solution, which is continuous piecewise linear, both close to the boundary and in the interior of the domain.

Michael Neilan, Abner J. Salgado, Wujun Zhang

We review the construction and analysis of numerical methods for strongly nonlinear PDEs, with an emphasis on convex and nonconvex fully nonlinear equations and the convergence to viscosity solutions. We begin by describing a fundamental result in this area which states that stable, consistent, and monotone schemes converge as the discretization parameter tends to zero. We review methodologies to construct finite difference, finite element, and semi-Lagrangian schemes that satisfy these criteria, and, in addition, discuss some rather novel tools that have paved the way to derive rates of convergence within this framework.

Ricardo H. Nochetto, Dimitrios Ntogkas, Wujun Zhang

In this paper we continue the analysis of the two-scale method for the Monge-Ampère equation for dimension $d \geq 2$ introduced in [10]. We prove continuous dependence of discrete solutions on data that in turn hinges on a discrete version of the Alexandroff estimate. They are both instrumental to prove pointwise error estimates for classical solutions with Hölder and Sobolev regularity. We also derive convergence rates for viscosity solutions with bounded Hessians which may be piecewise smooth or degenerate.

Abner J. Salgado, Wujun Zhang

We propose and analyze a two-scale finite element method for the Isaacs equation. The fine scale is given by the mesh size $h$ whereas the coarse scale $\varepsilon$ is dictated by an integro-differential approximation of the partial differential equation. We show that the method satisfies the discrete maximum principle provided that the mesh is weakly acute. This, in conjunction with weak operator consistency of the finite element method, allows us to establish convergence of the numerical solution to the viscosity solution as $\varepsilon, h\to0$, and $\varepsilon \gtrsim h^{1/2}|\log h|$. In addition, using a discrete Alexandrov Bakelman Pucci estimate we deduce rates of convergence, under suitable smoothness assumptions on the exact solution.

Michael Neilan, Wujun Zhang

We develop discrete $W^2_p$-norm error estimates for the Oliker-Prussner method applied to the Monge-Ampère equation. This is obtained by extending discrete Alexandroff estimates and showing that the contact set of a nodal function contains information on its second order difference. In addition, we show that the size of the complement of the contact set is controlled by the consistency of the method. Combining both observations, we show that the error estimate $\|u - u_h\|_{W^2_p} \leq C h^{1/p}$ if $p > d$ and $\|u - u_h\|_{W^2_p} \leq C h^{1/d} \big(\ln\left(\frac 1 h \right)\big)^{1/d} $ if $p \leq d$. Here the constant $C$ depends on $\|{u}\|_{C^{3,1}(\barΩ)}$, the dimension $d$, and the constant $p$. Numerical examples are given in two space dimensions and confirm that the estimate is sharp in several cases.

Ricardo H. Nochetto, Wujun Zhang

We study the Oliker-Prussner method exploiting its geometric nature. We derive discrete stability and continuous dependence estimates in the max-norm by using a discrete Alexandroff estimate and the Brunn-Minkowski inequality. We show that the method is exact for all convex quadratic polynomials provided the underlying set of nodes is translation invariant within the domain; nodes still conform to the domain boundary. This gives a suitable notion of operator consistency which, combined with stability, leads to pointwise rates of convergence for classical and non-classical solutions of the Monge-Ampère equation.

Ricardo H. Nochetto, Wujun Zhang

We design a two-scale finite element method (FEM) for linear elliptic PDEs in non-divergence form $A(x) : D^2 u(x) = f(x)$ in a bounded but not necessarily convex domain $Ω$ and study it in the max norm. The fine scale is given by the meshsize $h$ whereas the coarse scale $ε$ is dictated by an integro-differential approximation of the PDE. We show that the FEM satisfies the discrete maximum principle (DMP) for any uniformly positive definite matrix $A$ provided that the mesh is face weakly acute. We establish a discrete Alexandroff-Bakelman-Pucci (ABP) estimate which is suitable for finite element analysis. Its proof relies on a discrete Alexandroff estimate which expresses the min of a convex piecewise linear function in terms of the measure of its sub-differential, and thus of jumps of its gradient. The discrete ABP estimate leads, under suitable regularity assumptions on $A$ and $u$, to pointwise error estimates of the form \begin{equation*} \| u - u^ε_h \|_{L_\infty(Ω)} \leq \, C(A,u) \, h^{2α/(2 + α)} \big| \ln h \big| \qquad 0< α\leq 2, \end{equation*} provided $ε\approx h^{2/(2+α)}$. Such a convergence rate is at best of order $ h \big| \ln h \big|$, which turns out to be quasi-optimal.

Ricardo H. Nochetto, Shawn W. Walker, Wujun Zhang

We consider the simplest one-constant model, put forward by J. Ericksen, for nematic liquid crystals with variable degree of orientation. The equilibrium state is described by a director field $\mathbf{n}$ and its degree of orientation $s$, where the pair $(s, \mathbf{n})$ minimizes a sum of Frank-like energies and a double well potential. In particular, the Euler-Lagrange equations for the minimizer contain a degenerate elliptic equation for $\mathbf{n}$, which allows for line and plane defects to have finite energy. We present a structure preserving discretization of the liquid crystal energy with piecewise linear finite elements that can handle the degenerate elliptic part without regularization, and show that it is consistent and stable. We prove $Γ$-convergence of discrete global minimizers to continuous ones as the mesh size goes to zero. We develop a quasi-gradient flow scheme for computing discrete equilibrium solutions and prove it has a strictly monotone energy decreasing property. We present simulations in two and three dimensions to illustrate the method's ability to handle non-trivial defects. A music video summary of the paper is available on YouTube: "Mathematical Modeling and Simulation of Nematic Liquid Crystals (A Montage)," http://www.youtube.com/watch?v=pWWw7\_6cQ-U

Ricardo H. Nochetto, Shawn W. Walker, Wujun Zhang

We present a robust discretization of the Ericksen model of liquid crystals with variable degree of orientation coupled with colloidal effects and electric fields. The total energy consists of the Ericksen energy, a weak anchoring (or penalized Dirichlet) energy to model colloids, and an electrical energy for a given electric field. We describe our special discretization of the total energy along with a method to compute minimizers via a discrete quasi-gradient flow algorithm which has a strictly monotone energy decreasing property. Numerical experiments are given in two and three dimensions to illustrate that the method is able to capture non-trivial defect patterns, such as the Saturn ring defect. We conclude with a rigorous proof of the Gamma-convergence of our discrete energy to the continuous energy.