scholarly journals Multiblock Mortar Mixed Approach for Second Order Parabolic Problems

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 325 ◽  
Author(s):  
Muhammad Arshad ◽  
Madiha Sana ◽  
Muhammad Mustahsan
Keyword(s):  
Convergence Rates ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Second Order ◽  
Parabolic Problems ◽  
Optimal Order ◽  
Discrete Problem ◽  
Simulation Domain ◽  
Local Problems ◽  
Mixed Approach

In this paper, the multiblock mortar mixed approximation of second order parabolic partial differential equations is considered. In this method, the simulation domain is decomposed into the non-overlapping subdomains (blocks), and a physically-meaningful boundary condition is set on the mortar interface between the blocks. The governing equations hold locally on each subdomain region. The local problems on blocks are coupled by introducing a special approximation space on the interfaces of neighboring subdomains. Each block is locally covered by an independent grid and the standard mixed finite element method is applied to solve the local problem. The unique solvability of the discrete problem is shown, and optimal order convergence rates are established for the approximate velocity and pressure on the subdomain. Furthermore, an error estimate for the interface pressure in mortar space is presented. The numerical experiments are presented to validate the efficiency of the method.

Error Estimates and Iterative Procedure for Mixed Finite Element Solution of Second-Order Quasi-Linear Elliptic Problems

2004 ◽  
Vol 4 (4) ◽  
pp. 445-463 ◽  
Author(s):  
Mikhail Karchevsky ◽  
Alexander Fedotov
Keyword(s):  
Finite Element ◽  
Error Estimates ◽  
Elliptic Equations ◽  
Iterative Procedure ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Second Order ◽  
Discrete Problem ◽  
Arbitrary Power ◽  
Power Rate

AbstractThe mixed finite element method for second-order quasi-linear elliptic equations with nonlinearities of arbitrary power rate of growth is considered. Error estimates are obtained. An iterative method for corresponding discrete problem is proposed and investigated.

A NEW NONCONFORMING MIXED FINITE ELEMENT METHOD FOR LINEAR ELASTICITY

2006 ◽  
Vol 16 (07) ◽  
pp. 979-999 ◽  
Author(s):  
SON-YOUNG YI
Keyword(s):  
Finite Element ◽  
Linear Elasticity ◽  
Convergence Rates ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Optimal Order ◽  
Displacement Boundary Condition ◽  
Displacement Boundary ◽  
Optimal Convergence Rates ◽  
Optimal Order Convergence

We have developed new nonconforming mixed finite element methods for linear elasticity with a pure traction (displacement) boundary condition based on the Hellinger–Reissner variational principle using rectangular elements. Convergence analysis yields an optimal (suboptimal) convergence rate of [Formula: see text] for the L2-error of the stress and [Formula: see text] for the displacement in the pure traction (displacement) boundary problem. However, numerical experiments have yielded optimal-order convergence rates for both stress and displacement in both problems and have shown superconvergence for the displacement at the midpoint of each element. Moreover, we observed that the optimal convergence rates are still valid for large λ.

MIXED CONFORMING ELEMENTS FOR THE LARGE-BODY LIMIT IN MICROMAGNETICS

2013 ◽  
Vol 24 (01) ◽  
pp. 113-144 ◽  
Author(s):  
MARKUS AURADA ◽  
JENS M. MELENK ◽  
DIRK PRAETORIUS
Keyword(s):  
Mixed Finite Element ◽  
Large Body ◽  
Mixed Finite Element Method ◽  
Adaptive Strategy ◽  
Posteriori Error ◽  
Higher Order ◽  
Three Dimensions ◽  
Macroscopic Model ◽  
Error Estimator ◽  
Discrete Problem

We introduce a stabilized conforming mixed finite element method for a macroscopic model in micromagnetics. We show well-posedness of the discrete problem for higher order elements in two and three dimensions, develop a full a priori analysis for lowest order elements, and discuss the extension of the method to higher order elements. We introduce a residual-based a posteriori error estimator and present an adaptive strategy. Numerical examples illustrate the performance of the method.

scholarly journals Mixed Weak Galerkin Method for Heat Equation with Random Initial Condition

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Chenguang Zhou ◽  
Yongkui Zou ◽  
Shimin Chai ◽  
Fengshan Zhang
Keyword(s):  
Finite Element ◽  
Heat Equation ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Optimal Order ◽  
Random Initial Condition ◽  
Initial Condition ◽  
Polynomial Functions ◽  
Weak Galerkin ◽  
Convergence Estimates

This paper is devoted to the numerical analysis of weak Galerkin mixed finite element method (WGMFEM) for solving a heat equation with random initial condition. To set up the finite element spaces, we choose piecewise continuous polynomial functions of degree j+1 with j≥0 for the primary variables and piecewise discontinuous vector-valued polynomial functions of degree j for the flux ones. We further establish the stability analysis of both semidiscrete and fully discrete WGMFE schemes. In addition, we prove the optimal order convergence estimates in L2 norm for scalar solutions and triple-bar norm for vector solutions and statistical variance-type convergence estimates. Ultimately, we provide a few numerical experiments to illustrate the efficiency of the proposed schemes and theoretical analysis.

scholarly journals Numerical Analysis of anH1-Galerkin Mixed Finite Element Method for Time Fractional Telegraph Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Jinfeng Wang ◽  
Meng Zhao ◽  
Min Zhang ◽  
Yang Liu ◽  
Hong Li
Keyword(s):  
Finite Element ◽  
Mixed Finite Element ◽  
Finite Difference Methods ◽  
Mixed Finite Element Method ◽  
Telegraph Equation ◽  
Optimal Order ◽  
Optimal Order Of Convergence ◽  
Spatial Direction ◽  
Coupled Equations ◽  
Fractional Telegraph Equation

We discuss and analyze anH1-Galerkin mixed finite element (H1-GMFE) method to look for the numerical solution of time fractional telegraph equation. We introduce an auxiliary variable to reduce the original equation into lower-order coupled equations and then formulate anH1-GMFE scheme with two important variables. We discretize the Caputo time fractional derivatives using the finite difference methods and approximate the spatial direction by applying theH1-GMFE method. Based on the discussion on the theoretical error analysis inL2-norm for the scalar unknown and its gradient in one dimensional case, we obtain the optimal order of convergence in space-time direction. Further, we also derive the optimal error results for the scalar unknown inH1-norm. Moreover, we derive and analyze the stability ofH1-GMFE scheme and give the results of a priori error estimates in two- or three-dimensional cases. In order to verify our theoretical analysis, we give some results of numerical calculation by using the Matlab procedure.

scholarly journals A New Characteristic Nonconforming Mixed Finite Element Scheme for Convection-Dominated Diffusion Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Dongyang Shi ◽  
Qili Tang ◽  
Yadong Zhang
Keyword(s):  
Finite Element ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Auxiliary Variable ◽  
Diffusion Problem ◽  
Optimal Order ◽  
Order Error ◽  
Elliptic Projection ◽  
Indispensable Tool ◽  
Convection Dominated

A characteristic nonconforming mixed finite element method (MFEM) is proposed for the convection-dominated diffusion problem based on a new mixed variational formulation. The optimal order error estimates for both the original variableuand the auxiliary variableσwith respect to the space are obtained by employing some typical characters of the interpolation operator instead of the mixed (or expanded mixed) elliptic projection which is an indispensable tool in the traditional MFEM analysis. At last, we give some numerical results to confirm the theoretical analysis.

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