Dr. Junping Wang, NSF
Location
Sondheim Hall : 108
Date & Time
September 18, 2015, 11:30 am – 12:30 pm
Description
Title: Basic Principles of Weak Galerkin Finite Element Methods for PDEs
Abstract: This talk shall introduce a new numerical technique, called the weak Galerkin finite element method (WG), for partial differential equations. Weak Galerkin is a finite element method for PDEs where the differential operators (e.g., gradient, divergence, curl, Laplacian etc) in the variational forms are approximated by weak forms as generalized distributions. The WG discretization procedure often involves the solution of inexpensive problems defined locally on each element. The solution from the local problems can be regarded as a reconstruction of the corresponding differential operators. The fundamental difference between the weak Galerkin finite element method and other existing methods is the use of weak functions and weak derivatives (i.e., locally reconstructed differential operators) in the design of numerical schemes based on existing variational forms for the underlying PDE problems. Weak Galerkin is, therefore, a natural extension of the conforming Galerkin finite element method. Due to its great structural flexibility, the weak Galerkin finite element method is well suited to most partial differential equations by providing the needed stability and accuracy in approximation.
The presentation will start with the second order elliptic equation, for which WG shall be applied and explained in detail. In particular, the concept of weak gradient will be introduced and discussed for its role in the design of weak Galerkin finite element schemes. The speaker will then introduce a general notion of weak differential operators, such as weak Hessian, weak divergence, and weak curl etc. These weak differential operators shall serve as building blocks for WG finite element methods for other class of partial differential equations, such as the Stokes equation, the biharmonic equation, the Maxwell equations in electron magnetics theory, div-curl systems, and PDEs in non-divergence form (such as the Fokker-Planck equation). The speaker will demonstrate how WG can be applied to each of the applications, with a discussion on the main features. Furthermore, a mathematical convergence theory shall be briefly given for some applications. The talk should be accessible to graduate students with adequate training in computational methods.
Abstract: This talk shall introduce a new numerical technique, called the weak Galerkin finite element method (WG), for partial differential equations. Weak Galerkin is a finite element method for PDEs where the differential operators (e.g., gradient, divergence, curl, Laplacian etc) in the variational forms are approximated by weak forms as generalized distributions. The WG discretization procedure often involves the solution of inexpensive problems defined locally on each element. The solution from the local problems can be regarded as a reconstruction of the corresponding differential operators. The fundamental difference between the weak Galerkin finite element method and other existing methods is the use of weak functions and weak derivatives (i.e., locally reconstructed differential operators) in the design of numerical schemes based on existing variational forms for the underlying PDE problems. Weak Galerkin is, therefore, a natural extension of the conforming Galerkin finite element method. Due to its great structural flexibility, the weak Galerkin finite element method is well suited to most partial differential equations by providing the needed stability and accuracy in approximation.
The presentation will start with the second order elliptic equation, for which WG shall be applied and explained in detail. In particular, the concept of weak gradient will be introduced and discussed for its role in the design of weak Galerkin finite element schemes. The speaker will then introduce a general notion of weak differential operators, such as weak Hessian, weak divergence, and weak curl etc. These weak differential operators shall serve as building blocks for WG finite element methods for other class of partial differential equations, such as the Stokes equation, the biharmonic equation, the Maxwell equations in electron magnetics theory, div-curl systems, and PDEs in non-divergence form (such as the Fokker-Planck equation). The speaker will demonstrate how WG can be applied to each of the applications, with a discussion on the main features. Furthermore, a mathematical convergence theory shall be briefly given for some applications. The talk should be accessible to graduate students with adequate training in computational methods.
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