Error Analysis for Characteristics-Based Methods for Degenerate Parabolic Problems

Chen, Zhangxin; Ewing, Richard E.; Jiang, Qiaoyuan; Spagnuolo, Anna Maria

doi:10.1137/s003614290037068x

Cited by 38 publications

(23 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As numerical results shown in [18], these estimates are indeed sharp under (A7) and the present regularity assumption (2.3) for the problem under consideration.…”

Section: Error Estimates Isupporting

confidence: 73%

Degenerate two-phase incompressible flow

Chen

Ewing

2001

Numerische Mathematik

Self Cite

View full text Add to dashboard Cite

This is the third paper of a series in which we analyze mathematical properties and develop numerical methods for a degenerate ellipticparabolic partial differential system which describes the flow of two incompressible, immiscible fluids in porous media. In this paper we consider a finite element approximation for this system. The elliptic equation for the pressure and velocity is approximated by a mixed finite element method, while the degenerate parabolic equation for the saturation is approximated by a Galerkin finite element method. A fully discrete approximation is analyzed. Sharp error estimates in energy norms are obtained for this approximation. The error analysis does not use any regularization of the saturation equation; the error estimates are derived directly from the degenerate equation. Also, the analysis does not impose any restriction on the nature of degeneracy. Finally, it respects the minimal regularity on the solution of the differential system. Subject Classification (1991): 35K60, 35K65, 76S05, 76T05 Mathematics

show abstract

“…As numerical results shown in [18], these estimates are indeed sharp under (A7) and the present regularity assumption (2.3) for the problem under consideration.…”

Section: Error Estimates Isupporting

confidence: 73%

Degenerate two-phase incompressible flow

Chen

Ewing

2001

Numerische Mathematik

Self Cite

View full text Add to dashboard Cite

show abstract

“…In [15,16], we give the secondorder MMOC combine with the defect-correction and Variational Multiscale method for the Navier-Stokes equations. Some other characteristics methods were shown in [17][18][19][20][21][22][23].…”

Section: Introductionmentioning

confidence: 99%

Modified characteristics mixed defect-correction finite element method for the time-dependent Navier–Stokes problems

2014

Applicable Analysis

View full text Add to dashboard Cite

In this paper, a modified method of characteristics mixed defect-correction finite element method (MMOCMDCFEM) for the time-dependent Navier-Stokes problems, which is led by combining the characteristics time discretization with the two-step defect-correction in space, is presented. In this method, the hyperbolic part (the temporal and advection term) are treated by a characteristic tracking scheme. Then, we solve the equations with an added artificial viscosity term and correct these solutions by using a defect-correction technique. The theory analysis shows that this method has a good convergence property. In order to show the efficiency of the MMOCMDCFEM, we first present some numerical results of analytical solution problems. Then, we give some numerical results of lid-driven cavity flow with Re = 5000 and 7500. We also give some numerical results of flow around a cylinder problems, and compare the results with MMOCFEM's. The numerical results show that this method is highly Efficient.

show abstract

“…In [10], Chen give a characteristic mixed discontinuous finite element methods for advection-dominated diffusion problems. Some other characteristics methods were shown in [1,4,6,11,14,45,49].…”

Section: Introductionmentioning

confidence: 99%

Second order modified method of characteristics mixed defect-correction finite element method for time dependent Navier–Stokes problems

2011

Numer Algor

View full text Add to dashboard Cite

In this paper, a second order modified method of characteristics defect-correction (SOMMOCDC) mixed finite element method for the time dependent Navier-Stokes problems is presented. In this method, the hyperbolic part (the temporal and advection term) are treated by a second order characteristics tracking scheme, and the non-linear term is linearized at the same time. Then, we solve the equations with an added artificial viscosity term and correct this solution by using the defect-correction technique. The error analysis shows that this method has a good convergence property. In order to show the efficiency of the SOMMOCDC mixed finite element method, we first present some numerical results of an analytical solution problem, which agrees very well with our theoretical results. Then, we give some numerical results of lid-driven cavity flow with the Reynolds number Re = 5,000, 7,500 and 10,000. From these numerical results, we can see that the schemes can result in good accuracy, which shows that this method is highly efficient.Keywords Second order modified method of characteristics · Defect-correction finite element method · Time-dependent Navier-Stokes problems · Error estimate · Lid-driven problem · Incompressible flow Mathematics Subject Classification (2010) 76D05 · 76M10 · 65M60 · 65M12

show abstract

Error Analysis for Characteristics-Based Methods for Degenerate Parabolic Problems

Cited by 38 publications

References 31 publications

Degenerate two-phase incompressible flow

Degenerate two-phase incompressible flow

Modified characteristics mixed defect-correction finite element method for the time-dependent Navier–Stokes problems

Second order modified method of characteristics mixed defect-correction finite element method for time dependent Navier–Stokes problems

Contact Info

Product

Resources

About