Wujun Zhang

A Posteriori Error Estimates for HDG Methods

We present the first a posteriori error analysis of the so-called hybridizable discontinuous Galerkin (HDG) methods for second-order elliptic problems. We show that the error in the flux can be controlled by only two terms. The first term captures the so-called data oscillation. The second solely depends on the difference between the trace of the scalar approximation and the corresponding numerical trace. Numerical experiments verifying the reliability and efficiency of the estimate in two-space dimensions are presented.

A Posteriori Error Analysis for Hybridizable Discontinuous Galerkin Methods for Second Order Elliptic Problems

We present a unified a posteriori error analysis for hybridizable discontinuous Galerkin methods for second order elliptic partial differential equations. The main feature of this analysis is that it reduces the task of establishing the reliability and efficiency of the estimator to the verification of three simple conditions. This approach allows us not only to derive new estimates for hybridizable discontinuous Galerkin methods but also to recover well-known a posteriori error estimates for mixed methods and for the continuous Galerkin method.

Discrete ABP Estimate and Convergence Rates for Linear Elliptic Equations in Non-divergence Form

We design a two-scale finite element method (FEM) for linear elliptic PDEs in non-divergence form

Contraction property of adaptive hybridizable discontinuous galerkin methods

A(x) : D^2 u(x) = f(x)

Pointwise rates of convergence for the Oliker–Prussner method for the Monge–Ampère equation

A(x):D2u(x)=f(x) in a bounded but not necessarily convex domain

Two-scale method for the Monge-Ampère equation: Convergence to the viscosity solution

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