scholarly journals ANISOTROPIC QUASI-WILSON ELEMENT WITH CONFORMING FINITE ELEMENT APPROXIMATION FOR COUPLED CONTINUUM PIPE-FLOW/DARCY MODEL IN KARST AQUIFERS

2016 ◽  
Vol 21 (4) ◽  
pp. 431-449 ◽  
Author(s):  
Wei Liu ◽  
Jintao Cui
Keyword(s):  
Finite Element ◽  
Pipe Flow ◽  
Finite Element Approximation ◽  
Flow In Porous Media ◽  
Optimal Error Estimates ◽  
Element Approximation ◽  
Approximation Solution ◽  
Darcy Model ◽  
Conforming Finite Element ◽  
Wilson Element

This paper presents a numerical method for solving systems of partial differential equations describing flow in porous media with an embedded and inclined conduit pipe. This work considers a coupled continuum pipe-flow/Darcy model. The numerical schemes presented are based on combinations of the quasi-Wilson element on anisotropic mesh and the conforming finite element on regular mesh. The existence and uniqueness of the approximation solution are obtained. Optimal error estimates in both L2 and H1 norms are obtained independent of the regularity condition on the mesh. Numerical examples show the accuracy and efficiency of the proposed scheme.

NUMERICAL METHODS OF NEW MIXED FINITE ELEMENT SCHEME FOR SINGLE-PHASE COMPRESSIBLE FLOW

2013 ◽  
Vol 11 (01) ◽  
pp. 1350055 ◽  
Author(s):  
SHUYING ZHAI ◽  
XINLONG FENG ◽  
ZHIFENG WENG
Keyword(s):  
Finite Element ◽  
Compressible Flow ◽  
Single Phase ◽  
Mixed Finite Element ◽  
Finite Element Approximation ◽  
Optimal Error Estimates ◽  
Element Approximation ◽  
Finite Element Scheme ◽  
Element Scheme ◽  
Conforming Finite Element

In this paper, a new mixed finite element scheme is given based on the less regularity of velocity for the single phase compressible flow in practice. Based on the new mixed variational formulation, we give its stable conforming finite element approximation for the P0–P1 pair and its stabilized conforming finite element approximation for the P1–P1 pair. Moreover, optimal error estimates are derived in H1-norm and L2-norm for the approximation of pressure and error estimate in L2-norm for the approximation of velocity by using two methods. Finally, numerical tests confirm the theoretical results of our methods.

scholarly journals Well-posedness and H(div)-conforming finite element approximation of a linearised model for inviscid incompressible flow

2020 ◽  
Vol 30 (05) ◽  
pp. 847-865
Author(s):  
Gabriel Barrenechea ◽  
Erik Burman ◽  
Johnny Guzmán
Keyword(s):  
Finite Element ◽  
Error Estimates ◽  
Model Problem ◽  
Finite Element Approximation ◽  
Inviscid Flow ◽  
Element Approximation ◽  
Well Posedness ◽  
Hodge Laplacian ◽  
Conforming Finite Element ◽  
Velocity Approximation

We consider a linearised model of incompressible inviscid flow. Using a regularisation based on the Hodge Laplacian we prove existence and uniqueness of weak solutions for smooth domains. The model problem is then discretised using [Formula: see text](div)-conforming finite element methods, for which we prove error estimates for the velocity approximation in the [Formula: see text]-norm of order [Formula: see text]. We also prove error estimates for the pressure error in the [Formula: see text]-norm.

scholarly journals Conforming Finite Element Approximations for a Fourth-Order Steklov Eigenvalue Problem

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Hai Bi ◽  
Shixian Ren ◽  
Yidu Yang
Keyword(s):  
Finite Element ◽  
Eigenvalue Problem ◽  
Fourth Order ◽  
Finite Element Approximation ◽  
Continuous Operator ◽  
Element Approximation ◽  
Morley Element ◽  
Steklov Eigenvalue ◽  
Conforming Finite Element ◽  
Steklov Eigenvalue Problem

This paper characterizes the spectrum of a fourth-order Steklov eigenvalue problem by using the spectral theory of completely continuous operator. The conforming finite element approximation for this problem is analyzed, and the error estimate is given. Finally, the bounds for Steklov eigenvalues on the square domain are provided by Bogner-Fox-Schmit element and Morley element.

A prioriestimates and optimal finite element approximation of the MHD flow in smooth domains

2018 ◽  
Vol 52 (1) ◽  
pp. 181-206 ◽  
Author(s):  
Yinnian He ◽  
Jun Zou
Keyword(s):  
Magnetic Field ◽  
Finite Element ◽  
Error Estimates ◽  
Mhd Flow ◽  
Finite Element Approximation ◽  
Optimal Error Estimates ◽  
Element Approximation ◽  
Optimal Error ◽  
Discrete Velocity ◽  
Mhd System

We study a finite element approximation of the initial-boundary value problem of the 3D incompressible magnetohydrodynamic (MHD) system under smooth domains and data. We first establish several important regularities anda prioriestimates for the velocity, pressure and magnetic field (u,p,B) of the MHD system under the assumption that ∇u∈L4(0,T;L2(Ω)3 × 3) and ∇ ×B∈L4(0,T;L2(Ω)3). Then we formulate a finite element approximation of the MHD flow. Finally, we derive the optimal error estimates of the discrete velocity and magnetic field in energy-norm and the discrete pressure inL2-norm, and the optimal error estimates of the discrete velocity and magnetic field inL2-norm by means of a novel negative-norm technique, without the help of the standard duality argument for the Navier-Stokes equations.

A mixed-primal finite element approximation of a sedimentation–consolidation system

2016 ◽  
Vol 26 (05) ◽  
pp. 867-900 ◽  
Author(s):  
Mario Alvarez ◽  
Gabriel N. Gatica ◽  
Ricardo Ruiz-Baier
Keyword(s):  
Finite Element ◽  
Fixed Point ◽  
Volume Fraction ◽  
Variable Viscosity ◽  
Finite Element Approximation ◽  
Weak Form ◽  
Variational Problems ◽  
Optimal Error Estimates ◽  
Element Approximation ◽  
Coupled Flow

This paper is devoted to the mathematical and numerical analysis of a strongly coupled flow and transport system typically encountered in continuum-based models of sedimentation–consolidation processes. The model focuses on the steady-state regime of a solid–liquid suspension immersed in a viscous fluid within a permeable medium, and the governing equations consist in the Brinkman problem with variable viscosity, written in terms of Cauchy pseudo-stresses and bulk velocity of the mixture; coupled with a nonlinear advection — nonlinear diffusion equation describing the transport of the solids volume fraction. The variational formulation is based on an augmented mixed approach for the Brinkman problem and the usual primal weak form for the transport equation. Solvability of the coupled formulation is established by combining fixed point arguments, certain regularity assumptions, and some classical results concerning variational problems and Sobolev spaces. In turn, the resulting augmented mixed-primal Galerkin scheme employs Raviart–Thomas approximations of order [Formula: see text] for the stress and piecewise continuous polynomials of order [Formula: see text] for velocity and volume fraction, and its solvability is deduced by applying a fixed-point strategy as well. Then, suitable Strang-type inequalities are utilized to rigorously derive optimal error estimates in the natural norms. Finally, a few numerical tests illustrate the accuracy of the augmented mixed-primal finite element method, and the properties of the model.

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