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 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 On a Novel Numerical Scheme for Riesz Fractional Partial Differential Equations

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 2014
Author(s):  
Junjiang Lai ◽  
Hongyu Liu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Scheme ◽  
Numerical Solutions ◽  
A Priori ◽  
Time Derivative ◽  
Stability Estimates ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Sharp Stability

In this paper, we consider numerical solutions for Riesz space fractional partial differential equations with a second order time derivative. We propose a Galerkin finite element scheme for both the temporal and spatial discretizations. For the proposed numerical scheme, we derive sharp stability estimates as well as optimal a priori error estimates. Extensive numerical experiments are conducted to verify the promising features of the newly proposed method.

scholarly journals Error estimates of high-order numerical methods for solving time fractional partial differential equations

2018 ◽  
Vol 21 (3) ◽  
pp. 746-774 ◽  
Author(s):  
Zhiqiang Li ◽  
Yubin Yan
Keyword(s):  
Partial Differential Equations ◽  
Numerical Methods ◽  
Differential Equations ◽  
Error Estimates ◽  
High Order ◽  
Convergence Order ◽  
Fractional Partial Differential Equations ◽  
Step Size ◽  
Partial Differential ◽  
Method Error

Abstract Error estimates of some high-order numerical methods for solving time fractional partial differential equations are studied in this paper. We first provide the detailed error estimate of a high-order numerical method proposed recently by Li et al. [21] for solving time fractional partial differential equation. We prove that this method has the convergence order O(τ3−α) for all α ∈ (0, 1) when the first and second derivatives of the solution are vanish at t = 0, where τ is the time step size and α is the fractional order in the Caputo sense. We then introduce a new time discretization method for solving time fractional partial differential equations, which has no requirements for the initial values as imposed in Li et al. [21]. We show that this new method also has the convergence order O(τ3−α) for all α ∈ (0, 1). The proofs of the error estimates are based on the energy method developed recently by Lv and Xu [26]. We also consider the space discretization by using the finite element method. Error estimates with convergence order O(τ3−α + h2) are proved in the fully discrete case, where h is the space step size. Numerical examples in both one- and two-dimensional cases are given to show that the numerical results are consistent with the theoretical results.

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.

scholarly journals A finite element method for time fractional partial differential equations

2011 ◽  
Vol 14 (3) ◽  
Author(s):  
Neville Ford ◽  
Jingyu Xiao ◽  
Yubin Yan
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Fully Discrete ◽  
Convergence Error ◽  
Time Stepping ◽  
Element Method

AbstractIn this paper, we consider the finite element method for time fractional partial differential equations. The existence and uniqueness of the solutions are proved by using the Lax-Milgram Lemma. A time stepping method is introduced based on a quadrature formula approach. The fully discrete scheme is considered by using a finite element method and optimal convergence error estimates are obtained. The numerical examples at the end of the paper show that the experimental results are consistent with our theoretical results.

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