scholarly journals A Godunov-Mixed Finite Element Method on Changing Meshes for the Nonlinear Sobolev Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Tongjun Sun
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Optimal Error Estimates ◽  
Element Space ◽  
Diffusion Term ◽  
Sobolev Equations ◽  
Nonlinear Sobolev Equations ◽  
Element Method

A Godunov-mixed finite element method on changing meshes is presented to simulate the nonlinear Sobolev equations. The convection term of the nonlinear Sobolev equations is approximated by a Godunov-type procedure and the diffusion term by an expanded mixed finite element method. The method can simultaneously approximate the scalar unknown and the vector flux effectively, reducing the continuity of the finite element space. Almost optimal error estimates inL2-norm under very general changes in the mesh can be obtained. Finally, a numerical experiment is given to illustrate the efficiency of the method.

scholarly journals The Time DiscontinuousH1-Galerkin Mixed Finite Element Method for Linear Sobolev Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-10
Author(s):  
Hong Yu ◽  
Tongjun Sun ◽  
Na Li
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Approximate Solutions ◽  
Piecewise Polynomial ◽  
Sobolev Equations ◽  
Time Variables ◽  
Norm Error ◽  
Element Method

We combine theH1-Galerkin mixed finite element method with the time discontinuous Galerkin method to approximate linear Sobolev equations. The advantages of these two methods are fully utilized. The approximate schemes are established to get the approximate solutions by a piecewise polynomial of degree at mostq-1with the time variable. The existence and uniqueness of the solutions are proved, and the optimalH1-norm error estimates are derived. We get high accuracy for both the space and time variables.

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