A weak Galerkin finite element method for second-order elliptic problems

By Junping Wang, Xiu Ye. Journal of Computational and Applied Mathematics, 2011

683 citations recorded by the source corpus.

This paper introduces a finite element method by using a weakly defined gradient operator over generalized functions. The use of weak gradients and their approximations results in a new concept called discrete weak gradients which is expected to play an important role in numerical methods for partial differential equations. This article intends to provide a general framework for managing differential operators on generalized functions. As a demonstrative example, the discrete weak gradient operator is employed as a building block in the design of numerical schemes for a second order elliptic problem, in which the classical gradient operator is replaced by the discrete weak gradient. The resulting numerical scheme is called a weak Galerkin (WG) finite element method. It can be seen that the weak Galerkin method allows the use of totally discontinuous functions in the finite element procedure. For the second order elliptic problem, an optimal order error estimate in both a discrete H^1 and L^2 norms are established for the corresponding weak Galerkin finite element solutions. A superconvergence is also observed for the weak Galerkin approximation.

Subfields: Computational Mechanics, Electrical and Electronic Engineering, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/ea4f1702d28c86b8af171a0cd90f8b7ca810b1b6

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

1. Xiaoping Xie
2. Minfu Feng
3. Gang Chen
4. Junping Wang
5. Xiu Ye
6. Lin Mu
7. Yanqi Wang
8. Fang Wang

In Electrical and Electronic Engineering

1. Xiu Ye
2. Junping Wang
3. Lin Mu
4. Yanqi Wang
5. Fang Wang
6. Wenbin Chen
7. Xiaoping Xie
8. Minfu Feng

In Mechanics of Materials

1. Junping Wang
2. Xiu Ye
3. Lin Mu
4. Yanqi Wang
5. Fang Wang
6. Wenbin Chen
7. Xiaoping Xie
8. Minfu Feng

Citations in the corpus

• A weak Galerkin finite element method for the stokes equations (2013, methodology): “In [15] and [13], a weak gradient operator was introduced for scalar functions.”
• A Numerical Study on the Weak Galerkin Method for the Helmholtz Equation (2013, methodology): “Through rigorous error analysis, optimal order of convergence of the WG solution in both discrete H1 norm and L2 norm is established under minimum regularity assumptions in [33].”
• A new weak Galerkin finite element method for the Helmholtz equation (2015, methodology): “Weak Galerkin (WG) finite element methods are also nonconforming methods that were first introduced in [32] for second order elliptic equation.”
• Weak Galerkin method for the coupled Darcy-Stokes flow (2014, methodology): “Although the method is still very new, it has already demonstrated many nice properties in various cases [39, 30, 40, 31].”
• A weak Galerkin mixed finite element method for second order elliptic problems (2012, methodology): “The goal of this paper is to extend the weak Galerkin method of [22, 23, 29] to the variational formulation (1.”
• Robust Globally Divergence-Free Weak Galerkin Methods for Stokes Equations (2016, methodology): “The WG method was first proposed and analyzed to solve second-order elliptic problems [63,64].”