scholarly journals Discontinuous Galerkin finite element differential calculus and applications to numerical solutions of linear and nonlinear partial differential equations

2016 ◽  
Vol 299 ◽  
pp. 68-91 ◽  
Author(s):  
Xiaobing Feng ◽  
Thomas Lewis ◽  
Michael Neilan
Keyword(s):  
Finite Element ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Discontinuous Galerkin ◽  
Differential Calculus ◽  
Numerical Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Galerkin Finite Element ◽  
Discontinuous Galerkin Finite Element ◽  
Linear And Nonlinear

Continuous Solution of Partial Differential Equations on Non-Rectangular and Non-Continuous Geometry

2014 ◽  
Author(s):  
P. Venkataraman
Keyword(s):  
Finite Element ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Continuous Solution ◽  
Finite Difference Methods ◽  
Nonlinear Partial Differential Equations ◽  
Partial Differential ◽  
Mesh Free ◽  
Numeric Calculation ◽  
Linear And Nonlinear

A high order continuous solution is obtained for partial differential equations on non-rectangular and non-continuous domain using Bézier functions. This is a mesh free alternative to finite element or finite difference methods that are normally used to solve such problems. The problem is handled without any transformation and the setup is direct, simple, and involves minimizing the error in the residuals of the differential equations along with the error in the boundary conditions over the domain. The solution can be expressed in polynomial form. The effort is same for linear and nonlinear partial differential equations. The procedure is developed as a combination of symbolic and numeric calculation. The solution is obtained through the application of standard unconstrained optimization. A constrained approach is also developed for nonlinear partial differential equations. Examples include linear and nonlinear partial differential equations. The solution for linear partial differential equations is compared to finite element solutions from COMSOL.

scholarly journals Numerical Solutions of Steady Tensorial Transport Equations Using Discontinuous Galerkin Method Implemented in FreeFem++

2016 ◽  
Vol 8 (1) ◽  
pp. 29-39
Author(s):  
K. M. Helal
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Discontinuous Galerkin ◽  
Numerical Solutions ◽  
Transport Equations ◽  
Galerkin Finite Element Method ◽  
Galerkin Finite Element ◽  
Discontinuous Galerkin Finite Element ◽  
Reaction Equations ◽  
Element Method

AbstractThe main purpose of this paper is to approximate the solution of the steady tensorial transport equations using discontinuous Galerkin finite element method implemented with the finite element solver FreeFem++. After introducing the formulations of the tensorial transport equations, the analysis of its componentwise equations, i.e., advection-reaction equations have been discussed. Discretizing the transport problem using discontinuous Galerkin finite element method, the iterative fixed-point method is used to obtain the solutions. We present the numerical simulations of two-dimensional benchmark problem and observe the instability of elasticity. All the simulations are done using the script developed in FreeFem++.

Sign in / Sign up

Export Citation Format

Share Document