Superconvergence property of an over-penalized discontinuous Galerkin finite element gradient recovery method

Abstract

A polynomial preserving recovery method is introduced for over-penalized symmetric interior penalty discontinuous Galerkin solutions to a quasi-linear elliptic problem. As a post-processing method, the polynomial preserving recovery is superconvergent for the linear and quadratic elements under specified meshes in the regular and chevron patterns, as well as general meshes satisfying Condition ( , σ ). By means of the averaging technique, we prove the polynomial preserving recovery method for averaged solutions is superconvergent, satisfying similar estimates as those for conforming finite element methods. We deduce superconvergence of the recovered gradient directly from discontinuous solutions and naturally construct an a posteriori error estimator. Consequently, the a posteriori error estimator based on the recovered gradient is asymptotically exact. Extensive numerical results consistent with our analysis are presented.

Polynomial preserving recovery on boundary

“…Introduction. Gradient recovery [7,10,12,17,[20][21][22][23][24][25] is an effective and widely used post-processing technique in scientific and engineering computation. The main purpose of this techniques is to reconstruct a better numerical gradient from a finite element solution.…”

“…The main purpose of this techniques is to reconstruct a better numerical gradient from a finite element solution. It can be used for mesh smoothing, a posteriori error estimate [12,20,22,23,25], and adaptive finite element method even with anisotropic meshes [2,9,11,16]. More recently, the gradient recovery technique was applied to improve eigenvalue approximation as well [8,14,15,18].…”

A Fully Discrete SIPG Method for Solving Two Classes of Vortex Dominated Flows

“…In general, σ e shall be chosen sufficiently large to guarantee coercivity, more accurately, the threshold values of σ e in [22] are given for β ¼ 1 in the above formula, which is referred to an SIPG scheme. Especially, as β > 1, the scheme is referred to an over-penalized scheme and the threshold values of σ e are presented in [23,24]. Analogously, the SIPG discretization for (18) is given by…”

A Hybrid High-Order Method for a Class of Strongly Nonlinear Elliptic Boundary Value Problems

“…Bi et al [4,6,8,58] studied various a priori and a posteriori error estimates for (3). Recently, Gudi et al [47] analyzed the HHO finite element approximation for (3) and proved the existence of a local unique discrete solution using the Brouwer fixed point theorem and the contraction principle.…”