A stabilizer free weak Galerkin method for the Biharmonic Equation on Polytopal Meshes

A stabilizer free WG Method for the Stokes Equations with order two superconvergence on polytopal mesh

A stabilizer free WG method is introduced for the Stokes equations with superconvergence on polytopal mesh in primary velocity-pressure formulation. Convergence rates two order higher than the optimal-order for velocity of the WG approximation is proved in both an energy norm and the $L^2$ norm. Optimal order error estimate for pressure in the $L^2$ norm is also established. The numerical examples cover low and high order approximations, and 2D and 3D cases.

无凸性假设下多面体网格上双调和方程的自动稳定弱Galerkin有限元方法

Auto-Stabilized Weak Galerkin Finite Element Methods for Biharmonic Equations on Polytopal Meshes without Convexity Assumptions

提出一种带内置稳定子的自动稳定WG有限元方法，可在无凸性假设的多面体网格上求解双调和方程，支持任意维度与灵活多项式次数，并建立相应误差估计。

On stabilizer-free weak Galerkin finite element methods on polytopal meshes

A stabilizing/penalty term is often used in finite element methods with discontinuous approximations to enforce connection of discontinuous functions across element boundaries. Removing stabilizers from discontinuous finite element methods will simplify the formulations and reduce programming complexity significantly. The goal of this paper is to introduce a stabilizer free weak Galerkin finite element method for second order elliptic equations on polytopal meshes in 2D or 3D. This new WG method keeps a simple symmetric positive definite form and can work on polygonal/polyheral meshes. Optimal…

多面体网格上双调和方程的弱Galerkin有限元方法

Weak Galerkin Finite Element Methods for the Biharmonic Equation on Polytopal Meshes

针对原始形式双调和方程，提出并分析了一种新的弱Galerkin有限元方法，该方法在任意形状的多边形或多面体网格上使用间断分片多项式，具有高度鲁棒性和灵活性，并建立了离散H^2范数下的最优阶误差估计。

双调和方程的免稳定子弱伽辽金混合有限元方法

The stabilizer free weak Galerkin mixed finite elements method for the biharmonic equation

本文提出免稳定子弱伽辽金混合有限元方法求解双调和方程，通过投影算子建立误差方程并推导误差估计，数值实验验证了理论结果。

A stabilizer free, pressure robust, and superconvergence weak Galerkin finite element method for the Stokes Equations on polytopal mesh

In this paper, we propose a new stabilizer free and pressure robust WG method for the Stokes equations with super-convergence on polytopal mesh in the primary velocity-pressure formulation. Convergence rates with one order higher than the optimal-order for velocity in both energy-norm and the $L^2$-norm and for pressure in $L^2$-norm are proved in our proposed scheme. The $H$(div)-preserving operator has been constructed based on the polygonal mesh for arbitrary polynomial degrees and employed in the body source assembling to break the locking phenomenon induced by poor mass conservation in th…

使用降阶多项式求解双调和方程的无稳定子弱Galerkin有限元方法

The stabilizer-free weak Galerkin finite element method for the Biharmonic equation using polynomials of reduced order

针对双调和方程，研究通过降低弱函数边界多项式次数并修改弱laplacian定义改进SFWG方法，给出数值格式并获得$H^2$与$L^2$范数的最优阶误差估计，且经实验验证。

A stabilizer-free $C^0$ weak Galerkin method for the biharmonic equations

In this article, we present and analyze a stabilizer-free $C^0$ weak Galerkin (SF-C0WG) method for solving the biharmonic problem. The SF-C0WG method is formulated in terms of cell unknowns which are $C^0$ continuous piecewise polynomials of degree $k+2$ with $k\geq 0$ and in terms of face unknowns which are discontinuous piecewise polynomials of degree $k+1$. The formulation of this SF-C0WG method is without the stabilized or penalty term and is as simple as the $C^1$ conforming finite element scheme of the biharmonic problem. Optimal order error estimates in a discrete $H^2$-like norm and th…

一种用于非凸多边形网格上的Reissner-Mindlin板模型的简单弱Galerkin有限元方法

A Simple Weak Galerkin Finite Element Method for the Reissner-Mindlin Plate Model on Non-Convex Polytopal Meshes

本文提出了一种适用于非凸多边形网格的简单弱Galerkin有限元方法，用于Reissner-Mindlin板模型，简化了实现并扩展了应用范围。

使用弱Galerkin方法在一般多面体剖分上对Morley元的扩展

An Extension of the Morley Element on General Polytopal Partitions Using Weak Galerkin Methods

本文利用弱Galerkin方法将双调和方程的Morley元从三角元扩展到一般多面体元，通过局部构造弱导数设计数值格式，并建立了能量范数和$L^2$范数下的误差估计，最后通过数值实验验证了理论。

多面体网格上无凸性约束的自稳定弱Galerkin有限元方法

Auto-Stabilized Weak Galerkin Finite Element Methods on Polytopal Meshes without Convexity Constraints

本文提出一种带内置稳定器的自稳定弱Galerkin有限元方法，利用bubble函数将其扩展至凸与非凸单元，适用于任意维数，克服了以往限制，并在离散$H^1$和$L^2$范数下取得最优阶误差估计。

非凸多面体网格上 Stokes 方程的自动稳定弱 Galerkin 有限元方法

Auto-Stabilized Weak Galerkin Finite Element Methods for Stokes Equations on Non-Convex Polytopal Meshes

针对非凸多面体网格上的 Stokes 方程，提出以 bubble 函数实现自动稳定的弱 Galerkin 有限元方法，无需传统稳定项，并证明离散 $H^1$ 与 $L^2$ 范数下的最优阶误差估计。

A conforming discontinuous Galerkin finite element method for the Stokes problem on polytopal meshes

A new discontinuous Galerkin finite element method for the Stokes equations is developed in the primary velocity-pressure formulation. This method employs discontinuous polynomials for both velocity and pressure on general polygonal/polyhedral meshes. Most finite element methods with discontinuous approximation have one or more stabilizing terms for velocity and for pressure to guarantee stability and convergence. This new finite element method has the standard conforming finite element formulation, without any velocity or pressure stabilizers. Optimal-order error estimates are established for…

Stabilizer-Free Weak Galerkin Methods for Monotone Quasilinear Elliptic PDEs

In this paper, we study the stabilizer-free weak Galerkin methods on polytopal meshes for a class of second order elliptic boundary value problems of divergence form and with gradient nonlinearity in the principal coefficient. With certain assumptions on the nonlinear coefficient, we show that the discrete problem has a unique solution. This is achieved by showing that the associated operator satisfies certain continuity and monotonicity properties. With the help of these properties, we derive optimal error estimates in the energy norm. We present several numerical examples to verify the error…

自稳定弱伽辽金有限元方法用于非凸多面体网格上的Biot固结模型

Auto-Stabilized Weak Galerkin Finite Element Methods for Biot's consolidation model on Non-Convex Polytopal Meshes

本文提出了一种自稳定的弱伽辽金有限元方法，用于Biot固结模型的经典位移-压力双场公式。该方法在不使用传统稳定器的情况下实现了数值稳定性，通过弱伽辽金有限元进行空间离散化，并采用后向欧拉方案进行时间离散化，保证了完全隐式且稳定的公式。

曲面域上的有限元理论及其在DGFEMs中的应用

Finite element theory on curved domains with applications to DGFEMs

本文研究了曲面域上不连续有限元方法的稳定性与误差分析所用的关键估计，包括迹估计、反估计、离散Poincaré-Friedrichs不等式和最优插值估计，并证明了曲面单纯形的曲率界，通过分析泊松问题的IPDG方法和双调和问题的变种hp-DGFEM，验证了稳定性估计和最优先验误差估计。

非凸多面体网格上求解 Brinkman 方程的一种简单且稳健的弱 Galerkin 方法

A Simple and Robust Weak Galerkin Method for the Brinkman Equations on Non-Convex Polytopal Meshes

针对 Brinkman 方程在 Stokes 与 Darcy 主导区域间的统一稳定求解难题，提出无稳定子 WG 有限元方法，支持凸与非凸多面体网格，并证明最优阶误差估计。