arXiv subjects

Shuhao Cao

Publications and source records attributed to Shuhao Cao.

25 records · Page 2Linked to original sources

arXiv2021

A New Numerical Method for Div-Curl Systems with Low Regularity Assumptions

This paper presents a numerical method for div-curl systems with normal boundary conditions by using a finite element technique known as primal-dual weak Galerkin (PDWG). The PDWG finite element scheme for the div-curl system has two prominent features in that it offers not only an accurate and reliable numerical solution to the div-curl system under the low $H^α$-regularity ($α>0$) assumption for the true solution, but also an effective approximation of normal harmonic vector fields regardless the topology of the domain. Results of seven numerical experiments are presented to demonstrate the performance of the PDWG algorithm, including one example on the computation of discrete normal harmonic vector fields.

math.NA↗

arXiv2019

A Discretization-Accurate Stopping Criterion for Iterative Solvers for Finite Element Approximation

This paper introduces a discretization-accurate stopping criterion of symmetric iterative methods for solving systems of algebraic equations resulting from the finite element approximation. The stopping criterion consists of the evaluations of the discretization and the algebraic error estimators, that are based on the respective duality error estimator and the difference of two consecutive iterates. Iterations are terminated when the algebraic estimator is of the same magnitude as the discretization estimator. Numerical results for multigrid $V(1,1)$-cycle and symmetric Gauss-Seidel iterative methods are presented for the linear finite element approximation to the Poisson equations. A large reduction in computational cost is observed compared to the standard residual-based stopping criterion.

math.NA↗

arXiv2018

Anisotropic Error Estimates of The Linear Nonconforming Virtual Element Methods

A refined a priori error analysis of the lowest order (linear) nonconforming Virtual Element Method (VEM) for approximating a model Poisson problem is developed in both 2D and 3D. A set of new geometric assumptions is proposed on shape regularity of polytopal meshes. A new error equation for the lowest order (linear) nonconforming VEM is derived for any choice of stabilization, and a new stabilization using a projection on an extended element patch is introduced for the error analysis on anisotropic elements.

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arXiv2018

Anisotropic Error Estimates of The Linear Virtual Element Method on Polygonal Meshes

A refined a priori error analysis of the lowest order (linear) Virtual Element Method (VEM) is developed for approximating a model two dimensional Poisson problem. A set of new geometric assumptions is proposed on shape regularity of polygonal meshes. A new universal error equation for the lowest order (linear) VEM is derived for any choice of stabilization, and a new stabilization using broken half-seminorm is introduced to incorporate short edges naturally into the a priori error analysis on isotropic elements. The error analysis is then extended to a special class of anisotropic elements with high aspect ratio originating from a body-fitted mesh generator, which uses straight lines to cut a shape regular background mesh. Lastly, some commonly used tools for triangular elements are revisited for polygonal elements to give an in-depth view of these estimates' dependence on shapes.

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arXiv2018

A note on the error estimate of the virtual element methods

This short note reports a new derivation of the optimal order of the a priori error estimates for conforming virtual element methods (VEM) on 3D polyhedral meshes based on an error equation. The geometric assumptions, which are necessary for the optimal order of the conforming VEM error estimate in the $H^1$-seminorm, are relaxed for that in a bilinear form-induced energy norm.

math.NA↗

arXiv2016

Robust A Posteriori Error Estimation for Finite Element Approximation to H(curl) Problem

In this paper, we introduce a novel a posteriori error estimator for the conforming finite element approximation to the H(curl) problem with inhomogeneous media and with the right-hand side only in L^2. The estimator is of the recovery type. Independent with the current approximation to the primary variable (the electric field), an auxiliary variable (the magnetizing field) is recovered in parallel by solving a similar H(curl) problem. An alternate way of recovery is presented as well by localizing the error flux. The estimator is then defined as the sum of the modified element residual and the residual of the constitutive equation defining the auxiliary variable. It is proved that the estimator is approximately equal to the true error in the energy norm without the quasi-monotonicity assumption. Finally, we present numerical results for two H(curl) interface problems.

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arXiv2015

A Recovery-Based A Posteriori Error Estimator for H(curl) Interface Problems

This paper introduces a new recovery-based a posteriori error estimator for the lowest order Nedelec finite element approximation to the H(curl) interface problem. The error estimator is analyzed by establishing both the reliability and the efficiency bounds and is supported by numerical results. Under certain assumptions, it is proved that the reliability and efficiency constants are independent of the jumps of the coefficients.

math.NA↗