scholarly journals Efficient Temporal Third/Fourth-Order Finite Element Method for a Time-Fractional Mobile/Immobile Transport Equation with Smooth and Nonsmooth Data

Materials ◽  
2021 ◽  
Vol 14 (19) ◽  
pp. 5792
Author(s):  
Lijuan Nong ◽  
An Chen
Keyword(s):  
Finite Element ◽  
Transport Equation ◽  
Fourth Order ◽  
Diffusion Processes ◽  
Nonsmooth Data ◽  
Fully Discrete ◽  
Numerical Theory ◽  
Fractional Models ◽  
Finite Element Schemes ◽  
Physical Phenomena

In recent years, the numerical theory of fractional models has received more and more attention from researchers, due to the broad and important applications in materials and mechanics, anomalous diffusion processes and other physical phenomena. In this paper, we propose two efficient finite element schemes based on convolution quadrature for solving the time-fractional mobile/immobile transport equation with the smooth and nonsmooth data. In order to deal with the weak singularity of solution near t=0, we choose suitable corrections for the derived schemes to restore the third/fourth-order accuracy in time. Error estimates of the two fully discrete schemes are presented with respect to data regularity. Numerical examples are given to illustrate the effectiveness of the schemes.

scholarly journals Error estimates for approximations of distributed order time fractional diffusion with nonsmooth data

2016 ◽  
Vol 19 (1) ◽  
Author(s):  
Bangti Jin ◽  
Raytcho Lazarov ◽  
Dongwoo Sheen ◽  
Zhi Zhou
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Error Estimates ◽  
Diffusion Processes ◽  
Fractional Diffusion ◽  
Galerkin Finite Element Method ◽  
Nonsmooth Data ◽  
Galerkin Finite Element ◽  
Semidiscrete Scheme ◽  
Distributed Order

AbstractIn this work, we consider the numerical solution of a distributed order subdiffusion model, arising in the modeling of ultra-slow diffusion processes. We develop a space semidiscrete scheme based on the Galerkin finite element method, and establish error estimates optimal with respect to data regularity in

Finite Element Analysis of Maxwell’s Equations in Dispersive Lossy Bi-Isotropic Media

2013 ◽  
Vol 5 (04) ◽  
pp. 494-509 ◽  
Author(s):  
Yunqing Huang ◽  
Jichun Li ◽  
Yanping Lin
Keyword(s):  
Finite Element ◽  
Existence And Uniqueness ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Practical Implementation ◽  
Element Analysis ◽  
Fully Discrete ◽  
Isotropic Media ◽  
Discrete Finite Element ◽  
Finite Element Schemes

AbstractIn this paper, the time-dependent Maxwell’s equations used to modeling wave propagation in dispersive lossy bi-isotropic media are investigated. Existence and uniqueness of the modeling equations are proved. Two fully discrete finite element schemes are proposed, and their practical implementation and stability are discussed.

A numerical method for two-dimensional nonlinear modified time-fractional fourth-order diffusion equation

2019 ◽  
Vol 10 (01) ◽  
pp. 1941005 ◽  
Author(s):  
Haiyan He ◽  
Kaijie Liang ◽  
Baoli Yin
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Diffusion Equation ◽  
Fourth Order ◽  
Time Derivative ◽  
Second Order ◽  
Two Dimensional ◽  
The Finite Element Method ◽  
Discrete Scheme ◽  
Fully Discrete

In this paper, we consider the finite element method for two-dimensional nonlinear modified time-fractional fourth-order diffusion equation. In order to avoid using higher order elements, we introduce an intermediate variable [Formula: see text] and translate the fourth-order derivative of the original problem into a second-order coupled system. We discretize the fractional time derivative terms by using the [Formula: see text]-approximation and discretize the first-order time derivative term by using the second-order backward differentiation formula. In the fully discrete scheme, we implement the finite element method for the spatial approximation. Unconditional stability of the fully discrete scheme is proven and its optimal convergence order is obtained. Numerical experiments are carried out to demonstrate our theoretical analysis.

scholarly journals Error estimates for a robust finite element method of two-term time-fractional diffusion-wave equation with nonsmooth data

2021 ◽  
Author(s):  
Lijuan Nong ◽  
An Chen ◽  
Jianxiong Cao
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Wave Equation ◽  
Error Estimates ◽  
Piecewise Linear ◽  
Fractional Diffusion ◽  
Diffusion Wave ◽  
Nonsmooth Data ◽  
Fully Discrete ◽  
Element Method

In this paper, we consider a two-term time-fractional diffusion-wave equation which involves the fractional orders $\alpha\in (1, 2)$ and $\beta\in(0, 1)$, respectively. By using piecewise linear Galerkin finite element method in space and convolution quadrature based on second-order backward difference method in time, we obtain a robust fully discrete scheme. Error estimates for semidiscrete and fully discrete schemes are established with respect to nonsmooth data. Numerical experiments for two-dimensional problems are provided to illustrate the efficiency of the method and conform the theoretical results.

scholarly journals A fully discrete finite element scheme for the Derrida-Lebowitz-Speer-Spohn equation

2013 ◽  
Vol 9 (17) ◽  
pp. 97-110
Author(s):  
Jorge Mauricio Ruiz Vera ◽  
Ignacio Mantilla Prada
Keyword(s):  
Finite Element ◽  
Coupled System ◽  
Fully Discrete ◽  
Linear Evolution ◽  
Linear Evolution Equation ◽  
Interface Fluctuations ◽  
Quantum Semiconductor ◽  
Discrete Finite Element ◽  
Available Information ◽  
Physical Phenomena

The Derrida-Lebowitz-Speer-Spohn (DLSS) equation is a fourth order in space non-linear evolution equation. This equation arises in the study of interface fluctuations in spin systems and quantum semiconductor modelling. In this paper, we present a positive preserving finite element discrtization for a coupled-equation approach to the DLSS equation. Using the available information about the physical phenomena, we are able to set the corresponding boundary conditions for the coupled system. We prove existence of a global in time discrete solution by fixed point argument. Numerical results illustrate the quantum character of the equation. Finally a test of order of convergence of the proposed discretization scheme is presented.

Sign in / Sign up

Export Citation Format

Share Document