A mixed finite-element method for elliptic operators with Wentzell boundary condition

2018 ◽  
Vol 40 (1) ◽  
pp. 87-108
Author(s):  
Eberhard Bänsch ◽  
Markus Gahn
Keyword(s):  
Boundary Condition ◽  
Finite Element ◽  
Mixed Finite Element ◽  
A Priori ◽  
Mixed Finite Element Method ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Weak Formulation ◽  
Wentzell Boundary Condition ◽  
Polyhedral Domain

Abstract In this paper we introduce and analyze a mixed finite-element approach for a coupled bulk-surface problem of second order with a Wentzell boundary condition. The problem is formulated on a domain with a curved smooth boundary. We introduce a mixed formulation that is equivalent to the usual weak formulation. Furthermore, optimal a priori error estimates between the exact solution and the finite-element approximation are derived. To this end, the curved domain is approximated by a polyhedral domain introducing an additional geometrical error that has to be bounded. A computational result confirms the theoretical findings.

scholarly journals A mixed elasticity formulation for fluid-poroelastic structure interaction

2021 ◽  
Author(s):  
Tongtong Li ◽  
Ivan Yotov
Keyword(s):  
Finite Element ◽  
Stokes Equations ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Finite Element Approximation ◽  
Free Fluid ◽  
Physical Parameters ◽  
Element Approximation ◽  
Weak Formulation ◽  
Velocity Pressure

We develop a mixed finite element method for the coupled problem arising in the interaction between a free fluid governed by the Stokes equations and flow in deformable porous medium modeled by the Biot system of poroelasticity. Mass conservation, balance of stress, and the Beavers-Joseph-Saffman condition are imposed on the interface. We consider a fully mixed Biot formulation based on a weakly symmetric stress-displacement-rotation elasticity system and Darcy velocity-pressure flow formulation. A velocity-pressure formulation is used for the Stokes equations. The interface conditions are incorporated through the introduction of the traces of the structure velocity and the Darcy pressure as Lagrange multipliers. Existence and uniqueness of a solution are established for the continuous weak formulation. Stability and error estimates are derived for the semi-discrete continuous-in-time mixed finite element approximation. Numerical experiments are presented to verify the theoretical results and illustrate the robustness of the method with respect to the physical parameters.

scholarly journals AN H1 -GALERKIN MIXED FINITE ELEMENT APPROXIMATION OF A NONLOCAL HYPERBOLIC EQUATION

2017 ◽  
Vol 22 (5) ◽  
pp. 643-653
Author(s):  
Fengxin Chen ◽  
Zhaojie Zhou
Keyword(s):  
Finite Element ◽  
Hyperbolic Equation ◽  
Mixed Finite Element ◽  
A Priori ◽  
Finite Element Approximation ◽  
Nonlinear Hyperbolic Equation ◽  
Element Approximation ◽  
Discrete Scheme ◽  
Fully Discrete ◽  
Priori Error Estimates

In this paper we investigate a semi-discrete H1 -Galerkin mixed finite element approximation of one kind of nolocal second order nonlinear hyperbolic equation, which is often used to describe vibration of an elastic string. A priori error estimates for the semi-discrete scheme are derived. A fully discrete scheme is constructed and one numerical example is given to verify the theoretical findings.

scholarly journals Crank-Nicolson Fully DiscreteH1-Galerkin Mixed Finite Element Approximation of One Nonlinear Integrodifferential Model

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Fengxin Chen
Keyword(s):  
Finite Element ◽  
Mixed Finite Element ◽  
A Priori ◽  
Time Derivative ◽  
Finite Element Approximation ◽  
Optimal Order ◽  
Element Approximation ◽  
Fully Discrete ◽  
Priori Error Estimates ◽  
Crank Nicolson

We consider a fully discreteH1-Galerkin mixed finite element approximation of one nonlinear integrodifferential model which often arises in mathematical modeling of the process of a magnetic field penetrating into a substance. We adopt the Crank-Nicolson discretization for time derivative. Optimal order a priori error estimates for the unknown function inL2andH1norm and its gradient function inL2norm are presented. A numerical example is given to verify the theoretical results.

A Priori Error Estimates of Finite Element Methods for Linear Parabolic Integro-Differential Optimal Control Problems

2014 ◽  
Vol 6 (5) ◽  
pp. 552-569 ◽  
Author(s):  
Wanfang Shen ◽  
Liang Ge ◽  
Danping Yang ◽  
Wenbin Liu
Keyword(s):  
Optimal Control ◽  
Finite Element ◽  
Error Estimates ◽  
A Priori ◽  
Mathematical Formulation ◽  
Finite Element Approximation ◽  
A Priori Error Estimates ◽  
Element Approximation ◽  
Weak Formulation ◽  
Priori Error Estimates

AbstractIn this paper, we study the mathematical formulation for an optimal control problem governed by a linear parabolic integro-differential equation and present the optimality conditions. We then set up its weak formulation and the finite element approximation scheme. Based on these we derive the a priori error estimates for its finite element approximation both in H1 and L2 norms. Furthermore some numerical tests are presented to verify the theoretical results.

scholarly journals A sign preserving mixed finite element approximation for contact problems

2011 ◽  
Vol 21 (3) ◽  
pp. 487-498
Author(s):  
Patrick Hild
Keyword(s):  
Finite Element ◽  
Contact Pressure ◽  
Displacement Field ◽  
Mixed Finite Element ◽  
A Priori ◽  
Finite Element Approximation ◽  
Contact Problems ◽  
Standard Approach ◽  
Normal Displacement ◽  
Element Approximation

A sign preserving mixed finite element approximation for contact problemsThis paper is concerned with the frictionless unilateral contact problem (i.e., a Signorini problem with the elasticity operator). We consider a mixed finite element method in which the unknowns are the displacement field and the contact pressure. The particularity of the method is that it furnishes a normal displacement field and a contact pressure satisfying the sign conditions of the continuous problem. Thea priorierror analysis of the method is closely linked with the study of a specific positivity preserving operator of averaging type which differs from the one of Chen and Nochetto. We show that this method is convergent and satisfies the samea priorierror estimates as the standard approach in which the approximated contact pressure satisfies only a weak sign condition. Finally we perform some computations to illustrate and compare the sign preserving method with the standard approach.

scholarly journals Numerical analysis of the mixed finite element method for the neutron diffusion eigenproblem with heterogeneous coefficients

2018 ◽  
Vol 52 (5) ◽  
pp. 2003-2035
Author(s):  
P. Ciarlet ◽  
L. Giret ◽  
E. Jamelot ◽  
F.D. Kpadonou
Keyword(s):  
Finite Element ◽  
Domain Decomposition ◽  
Decomposition Method ◽  
Source Term ◽  
Domain Decomposition Method ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Different Characteristics

We study first the convergence of the finite element approximation of the mixed diffusion equations with a source term, in the case where the solution is of low regularity. Such a situation commonly arises in the presence of three or more intersecting material components with different characteristics. Then we focus on the approximation of the associated eigenvalue problem. We prove spectral correctness for this problem in the mixed setting. These studies are carried out without, and then with a domain decomposition method. The domain decomposition method can be non-matching in the sense that the traces of the finite element spaces may not fit at the interface between subdomains. Finally, numerical experiments illustrate the accuracy of the method.

The Numerical Analysis and Simulation of a Linearized Crank-Nicolson H1-GMFEM for Nonlinear Coupled BBM Equations

2014 ◽  
Vol 513-517 ◽  
pp. 1919-1926 ◽  
Author(s):  
Min Zhang ◽  
Zu Deng Yu ◽  
Yang Liu ◽  
Hong Li
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Scheme ◽  
Mixed Finite Element ◽  
A Priori ◽  
Numerical Test ◽  
Mixed Finite Element Method ◽  
Spatial Direction ◽  
Time Direction ◽  
Crank Nicolson

In this article, the numerical scheme of a linearized Crank-Nicolson (C-N) method based on H1-Galerkin mixed finite element method (H1-GMFEM) is studied and analyzed for nonlinear coupled BBM equations. In this method, the spatial direction is approximated by an H1-GMFEM and the time direction is discretized by a linearized Crank-Nicolson method. Some optimal a priori error results are derived for four important variables. For conforming the theoretical analysis, a numerical test is presented.

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