scholarly journals IMPROVED ACCURACY FOR ALTERNATING-DIRECTION METHODS FOR PARABOLIC EQUATIONS BASED ON REGULAR AND MIXED FINITE ELEMENTS

2007 ◽  
Vol 17 (08) ◽  
pp. 1279-1305 ◽  
Author(s):  
TODD ARBOGAST ◽  
CHIEH-SEN HUANG ◽  
SONG-MING YANG
Keyword(s):  
Finite Element ◽  
Mixed Finite Element ◽  
Mixed Finite Elements ◽  
Time Step ◽  
Galerkin Finite Element ◽  
Alternating Direction ◽  
Direction Splitting ◽  
Proposed Modification ◽  
Alternating Directions Method ◽  
Improved Accuracy

An efficient modification by Douglas and Kim of the usual alternating directions method reduces the splitting error from [Formula: see text] to [Formula: see text] in time step k. We prove convergence of this modified alternating directions procedure, for the usual non-mixed Galerkin finite element and finite difference cases, under the restriction that k/h2 is sufficiently small, where h is the grid spacing. This improves the results of Douglas and Gunn, who require k/h4 to be sufficiently small, and Douglas and Kim, who require that the locally one-dimensional operators commute. We propose a similar and efficient modification of alternating directions for mixed finite element methods that reduces the splitting error to [Formula: see text], and we prove convergence in the noncommuting case, provided that k/h2 is sufficiently small. Numerical computations illustrating the mixed finite element results are also presented. They show that our proposed modification can lead to a significant reduction in the alternating direction splitting error.

scholarly journals Domain Decomposition Modified with Characteristic Mixed Finite Element and Numerical Analysis for Three-Dimensional Slightly Compressible Oil-Water Seepage Displacement

2017 ◽  
Vol 9 (1) ◽  
pp. 143
Author(s):  
Yirang Yuan ◽  
Luo Chang ◽  
Changfeng Li ◽  
Tongjun Sun
Keyword(s):  
Finite Element ◽  
Domain Decomposition ◽  
Method Of Characteristics ◽  
Mixed Finite Element ◽  
Three Dimensional ◽  
Optimal Error Estimate ◽  
Computational Domain ◽  
Time Step ◽  
Slightly Compressible ◽  
The Method Of Characteristics

A parallel algorithm is presented to solve three-dimensional slightly compressible seepage displacement where domain decomposition and characteristics-mixed finite element are combined. Decomposing the computational domain into several subdomains, we define a special function to approximate the derivative at interior boundary explicitly and obtain numerical solutions of the saturation implicitly on subdomains in parallel. The method of characteristics can confirm strong stability at the fronts, and can avoid numerical dispersion and nonphysical oscillation. It can adopt large-time step but can obtain small time truncation error. So a characteristic domain decomposition finite element scheme is put forward to compute the saturation. The flow equation is computed by the method of mixed finite element and numerical accuracy of Darcy velocity is improved one order. For a model problem we apply some techniques such as variation form, domain decomposition, the method of characteristics, the principle of energy, negative norm estimates, induction hypothesis, and the theory of priori estimates of differential equations to derive optimal error estimate in $l^2$ norm. Numerical example is given to testify theoretical analysis and numerical data show that this method is effective in solving actual applications. Then it can solve the well-known problem.

scholarly journals Higher order multipoint flux mixed finite element methods on quadrilaterals and hexahedra

2019 ◽  
Vol 29 (06) ◽  
pp. 1037-1077 ◽  
Author(s):  
Ilona Ambartsumyan ◽  
Eldar Khattatov ◽  
Jeonghun J. Lee ◽  
Ivan Yotov
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Mixed Finite Element ◽  
Mixed Finite Elements ◽  
Higher Order ◽  
Local Velocity ◽  
New Family ◽  
Gauss Points ◽  
Theoretical Results ◽  
Optimal Formula

We develop higher order multipoint flux mixed finite element (MFMFE) methods for solving elliptic problems on quadrilateral and hexahedral grids that reduce to cell-based pressure systems. The methods are based on a new family of mixed finite elements, which are enhanced Raviart–Thomas spaces with bubbles that are curls of specially chosen polynomials. The velocity degrees of freedom of the new spaces can be associated with the points of tensor-product Gauss–Lobatto quadrature rules, which allows for local velocity elimination and leads to a symmetric and positive definite cell-based system for the pressures. We prove optimal [Formula: see text]th order convergence for the velocity and pressure in their natural norms, as well as [Formula: see text]st order superconvergence for the pressure at the Gauss points. Moreover, local postprocessing gives a pressure that is superconvergent of order [Formula: see text] in the full [Formula: see text]-norm. Numerical results illustrating the validity of our theoretical results are included.

scholarly journals Analysis of Subgrid Stabilization Method for Stokes-Darcy Problems

2016 ◽  
Vol 2016 ◽  
pp. 1-9
Author(s):  
Kamel Nafa
Keyword(s):  
Finite Element ◽  
Stokes Equations ◽  
Fluid Motion ◽  
Mixed Finite Elements ◽  
Point Of View ◽  
Darcy Law ◽  
Element Formulation ◽  
Computational Point ◽  
Galerkin Finite Element ◽  
Darcy Problem

A number of techniques, used as remedy to the instability of the Galerkin finite element formulation for Stokes like problems, are found in the literature. In this work we consider a coupled Stokes-Darcy problem, where in one part of the domain the fluid motion is described by Stokes equations and for the other part the fluid is in a porous medium and described by Darcy law and the conservation of mass. Such systems can be discretized by heterogeneous mixed finite elements in the two parts. A better method, from a computational point of view, consists in using a unified approach on both subdomains. Here, the coupled Stokes-Darcy problem is analyzed using equal-order velocity and pressure approximation combined with subgrid stabilization. We prove that the obtained finite element solution is stable and converges to the classical solution with optimal rates for both velocity and pressure.

DYNAMICAL FINITE ELEMENT MODELING OF SOFT TISSUES AS CHEMOELECTRIC POROUS MEDIA

2011 ◽  
Vol 11 (01) ◽  
pp. 101-130 ◽  
Author(s):  
ZHAOCHUN YANG ◽  
PATRICK SMOLINSKI ◽  
JEEN-SHANG LIN ◽  
LARS G. GILBERTSON
Keyword(s):  
Finite Element ◽  
Soft Tissues ◽  
Water Pressure ◽  
Mixed Finite Element ◽  
Viscoelastic Body ◽  
Biological Tissues ◽  
Element Formulation ◽  
Dynamical Process ◽  
Time Step ◽  
Charge Concentration

An implicit mixed finite element formulation of hydrated soft biological tissues, based on the Simon model, is presented that incorporates the coupling of solid, fluid, and ion phases as well as the viscoelasticity of soft tissue in the dynamical process. The tissues are modeled as a multi-field viscoelastic body subject to finite deformation. In addition to a three-field (u-w-p) modeling of the porous matrix, the study also includes an ion phase for the ionic solution. After presenting the formulation, an efficient staggered solution scheme is presented: within each time step, the ion charge equation is solved first to give the distribution of the charge concentration, the charge induced osmotic water pressure is then employed in solving the u-w-p equations. The resulting u field becomes a forcing term to the solution of the ion charge concentration equations for iteration. This methodology and codes developed for the study have been verified with one-dimensional (1D) analytical solutions. A 2D chemical electric swelling model illustrates the important role of viscoelasticity. A brain tissue impact example demonstrates the potential application of the model.

scholarly journals MIXED FINITE ELEMENT APPROXIMATION OF THE VECTOR LAPLACIAN WITH DIRICHLET BOUNDARY CONDITIONS

2012 ◽  
Vol 22 (09) ◽  
pp. 1250024 ◽  
Author(s):  
DOUGLAS N. ARNOLD ◽  
RICHARD S. FALK ◽  
JAY GOPALAKRISHNAN
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Stokes Equations ◽  
Mixed Finite Element ◽  
Mixed Finite Elements ◽  
Mixed Finite Element Method ◽  
Two Dimensions ◽  
Dirichlet Boundary Conditions ◽  
Element Approximation

We consider the finite element solution of the vector Laplace equation on a domain in two dimensions. For various choices of boundary conditions, it is known that a mixed finite element method, in which the rotation of the solution is introduced as a second unknown, is advantageous, and appropriate choices of mixed finite element spaces lead to a stable, optimally convergent discretization. However, the theory that leads to these conclusions does not apply to the case of Dirichlet boundary conditions, in which both components of the solution vanish on the boundary. We show, by computational example, that indeed such mixed finite elements do not perform optimally in this case, and we analyze the suboptimal convergence that does occur. As we indicate, these results have implications for the solution of the biharmonic equation and of the Stokes equations using a mixed formulation involving the vorticity.

scholarly journals A Framework for Nonconforming Mixed Finite Element Method for Elliptic Problems in ℝ3

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Gwanghyun Jo ◽  
J. H. Kim
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Experiments ◽  
Mixed Finite Element ◽  
Mixed Finite Elements ◽  
Mixed Finite Element Method ◽  
Optimal Order ◽  
Order Convergence ◽  
New Family ◽  
Optimal Order Convergence

In this paper, we suggest a new patch condition for nonconforming mixed finite elements (MFEs) on parallelepiped and provide a framework for the convergence. Also, we introduce a new family of nonconforming MFE space satisfying the new patch condition. The numerical experiments show that the new MFE shows optimal order convergence in Hdiv and L2-norm for various problems with discontinuous coefficient case.

scholarly journals Superconvergence of Semidiscrete Splitting Positive Definite Mixed Finite Elements for Hyperbolic Optimal Control Problems

2022 ◽  
Vol 2022 ◽  
pp. 1-10
Author(s):  
Yuchun Hua ◽  
Yuelong Tang
Keyword(s):  
Optimal Control ◽  
Finite Element ◽  
Optimal Control Problems ◽  
Hyperbolic Equations ◽  
Mixed Finite Element ◽  
Mixed Finite Elements ◽  
Positive Definite ◽  
The State ◽  
Control Problems ◽  
Convergence And Superconvergence

In this paper, we consider semidiscrete splitting positive definite mixed finite element methods for optimal control problems governed by hyperbolic equations with integral constraints. The state and costate are approximated by the lowest order Raviart-Thomas mixed rectangular finite element, and the control is approximated by piecewise constant functions. We derive some convergence and superconvergence results for the control, the state and the adjoint state. A numerical example is provided to demonstrate our theoretical results.

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