scholarly journals A New Expanded Mixed Element Method for Convection-Dominated Sobolev Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Jinfeng Wang ◽  
Yang Liu ◽  
Hong Li ◽  
Zhichao Fang
Keyword(s):  
Error Estimates ◽  
Existence And Uniqueness ◽  
A Priori ◽  
A Priori Error Estimates ◽  
Element Solution ◽  
Sobolev Equation ◽  
Priori Error Estimates ◽  
Mixed Element ◽  
Convection Dominated ◽  
Element Method

We propose and analyze a new expanded mixed element method, whose gradient belongs to the simple square integrable space instead of the classicalH(div; Ω) space of Chen’s expanded mixed element method. We study the new expanded mixed element method for convection-dominated Sobolev equation, prove the existence and uniqueness for finite element solution, and introduce a new expanded mixed projection. We derive the optimal a priori error estimates inL2-norm for the scalar unknownuand a priori error estimates in(L2)2-norm for its gradientλand its fluxσ. Moreover, we obtain the optimal a priori error estimates inH1-norm for the scalar unknownu. Finally, we obtained some numerical results to illustrate efficiency of the new method.

scholarly journals A New Mixed Element Method for a Class of Time-Fractional Partial Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Yang Liu ◽  
Hong Li ◽  
Wei Gao ◽  
Siriguleng He ◽  
Zhichao Fang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Error Estimates ◽  
A Priori ◽  
A Priori Error Estimates ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Priori Error Estimates ◽  
Mixed Element ◽  
Element Method

A kind of new mixed element method for time-fractional partial differential equations is studied. The Caputo-fractional derivative of time direction is approximated by two-step difference method and the spatial direction is discretized by a new mixed element method, whose gradient belongs to the simpleL2Ω2space replacing the complexH(div;Ω)space. Some a priori error estimates inL2-norm for the scalar unknownuand inL22-norm for its gradientσ. Moreover, we also discuss a priori error estimates inH1-norm for the scalar unknownu.

scholarly journals Bubbles Enriched Quadratic Finite Element Method for the 3D-Elliptic Obstacle Problem

2018 ◽  
Vol 18 (2) ◽  
pp. 223-236 ◽  
Author(s):  
Sharat Gaddam ◽  
Thirupathi Gudi
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Error Estimates ◽  
Obstacle Problem ◽  
A Priori ◽  
A Posteriori Error Estimates ◽  
Three Dimensional ◽  
A Priori Error Estimates ◽  
Priori Error Estimates ◽  
Element Method

AbstractAn optimally convergent (with respect to the regularity) quadratic finite element method for the two-dimensional obstacle problem on simplicial meshes is studied in [14]. There was no analogue of a quadratic finite element method on tetrahedron meshes for the three-dimensional obstacle problem. In this article, a quadratic finite element enriched with element-wise bubble functions is proposed for the three-dimensional elliptic obstacle problem. A priori error estimates are derived to show the optimal convergence of the method with respect to the regularity. Further, a posteriori error estimates are derived to design an adaptive mesh refinement algorithm. A numerical experiment illustrating the theoretical result on a priori error estimates is presented.

A NEW CHARACTERISTIC EXPANDED MIXED METHOD FOR SOBOLEV EQUATION WITH CONVECTION TERM

2013 ◽  
Vol 05 (01) ◽  
pp. 1350015 ◽  
Author(s):  
YANG LIU ◽  
HONG LI ◽  
SIRIGULENG HE ◽  
ZHICHAO FANG ◽  
JINFENG WANG
Keyword(s):  
Error Estimates ◽  
Mixed Method ◽  
A Priori ◽  
A Priori Error Estimates ◽  
Characteristic Method ◽  
Convection Term ◽  
Sobolev Equation ◽  
Diffusion Term ◽  
Mixed Scheme ◽  
Priori Error Estimates

In this paper, a new numerical method based on a new expanded mixed scheme and the characteristic method is developed and discussed for Sobolev equation with convection term. The hyperbolic part [Formula: see text] is handled by the characteristic method and the diffusion term ∇ ⋅ (a(x, t)∇u+b(x, t)∇ut) is approximated by the new expanded mixed method, whose gradient belongs to the simple square integrable (L2(Ω))2 space instead of the classical H( div ; Ω) space. For a priori error estimates, some important lemmas based on the new expanded mixed projection are introduced. An optimal priori error estimates in L2-norm for the scalar unknown u and a priori error estimates in (L2)2-norm for its gradient λ, and its flux σ (the coefficients times the negative gradient) are derived. In particular, an optimal priori error estimate in H1-norm for the scalar unknown u is obtained.

scholarly journals A note on a priori error estimates for augmented mixed methods

2016 ◽  
Vol 57 ◽  
pp. 139-144
Author(s):  
Tomás P. Barrios ◽  
Edwin Behrens ◽  
Rommel Bustinza
Keyword(s):  
Mixed Methods ◽  
Error Estimates ◽  
A Priori ◽  
A Priori Error Estimates ◽  
Priori Error Estimates
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