Preprints: Vincent J. Ervin

Fractional Advection Diffusion Equations

3. Ervin, V.J., Heuer, N. and Roop, J.P.,  “Numerical Approximation of a Time Dependent, Non-linear, Fractional
   Order Diffusion Equation,” SIAM J. Numer. Anal., 45, 572-591, (2007).

Abstract: We study  a defect correction method for the approximation of viscoelastic fluid flow. In the defect step, the constitutive equation is computed with an artificially reduced Weissenberg parameter for stability, and the resulting residual is corrected in the correction step.  We prove the convergence of the defect correction method and derive an error estimate for the Oseen-viscoelastic model problem. The derived theoretical results are supported by  numerical tests for both the Oseen-viscoelastic problem and the Johnson-Segalman model problem.
(paper in .pdf format)

2. Ervin, V.J., and Roop, J.P., “Variational Solution of Fractional Advection Dispersion Equations on Bounded Domains in R^{d},” Numer. Methods for Partial Differential Equations, 23, 256-281, (2007).

Abstract:   In this article we analyze a fully discrete numerical approximation to a time dependent fractional order diffusion equation which contains a non-local, quadratic non-linearity. The analysis is performed for a general fractional order diffusion operator. The non-linear term studied is a product of the unknown function and a convolution operator of order 0. Convergence of the approximation and a priori error estimates are given. Numerical computations are included which confirm the theoretical predictions.
(paper in .pdf format)

1. Ervin, V.J., and Roop, J.P., “Variational Formulation for the Stationary Fractional Advection Dispersion Equation,” Numer. Methods for Partial Differential Equations, 22, 558-576, (2006).

Abstract:  In this paper a theoretical framework for the Galerkin finite element approximation to the steady state fractional advection dispersion equation is presented. Appropriate fractional derivative spaces are defined and shown to be equivalent to the usual fractional dimension Sobolev spaces H^{s}. Existence and uniqueness results are proven, and error estimates for the Galerkin approximation derived. Numerical results are included which confirm the theoretical estimates.
(paper in .pdf format)

Viscoelasticity

14. Ervin, V.J., Jenkins, E.W., and Sun, S., “Coupled Generalized Non-linear Stokes Flow with flow through a Porous Media,” submitted to SIAM J. Numer. Anal.

13. Chrispell, J.C, Ervin, V.J., and Jenkins, E.W., “A Fractional Step Θ-method for Viscoelastic Fluid Flow using a SUPG approximation,” to appear International Journal of Computational Science.

12. Ervin, V.J., Howell, J.S. and Stanculescu, I., “A Dual-Mixed Approximation Method for a Three-field Model of a Non-linear Generalized Stokes Problem,” submitted to Comp. Meth. Appl. Mech. Eng.

11. Ervin, V.J., Howell, J.S. and Lee, H., “A Two-Parameter Defect-Correction Method for Computation of Steady-State Viscoelastic Fluid Flow,” to appear Appl. Math. Comput.

10. Chrispell, J.C, Ervin, V.J., and Jenkins, E.W., “A Fractional Step Θ-method for Convection-Diffusion Problems,” J. Math. Anal. Appl., 333, 204-218, (2007).

9. Ervin, V.J., and Lee, H., “Numerical Approximation of a quasi-Newtonian Stokes Flow Problem with Defective Boundary Conditions,” SIAM J. Numer. Anal., 45, 2120-2140, (2007).

8. Ervin, V.J., and Lee, H., “Defect Correction Method for Viscoelastic Fluid Flows at High Weissenberg Number,” Numer. Methods for Partial Differential Equations, 22, 145-164, (2006).

Abstract: We study  a defect correction method for the approximation of viscoelastic fluid flow. In the defect step, the constitutive equation is computed with an artificially reduced Weissenberg parameter for stability, and the resulting residual is corrected in the correction step.  We prove the convergence of the defect correction method and derive an error estimate for the Oseen-viscoelastic model problem. The derived theoretical results are supported by  numerical tests for both the Oseen-viscoelastic problem and the Johnson-Segalman model problem.
(paper in .pdf format)

7. Ervin, V.J., and Phillips, T.N., “Residual A Posteriori Error Estimator for a Three Field Model of a Generalized Stokes Problem,” Comp. Meth. Appl. Mech. Eng., 195, 2599-2610, (2006).

Abstract: In this article we propose and analyze an a posteriori error estimator for a three-field model of a generalized Stokes problem. The components of the a posteriori error estimator are defined via a non-linear projection of the residues of the variational equations.
Both upper and lower bounds for the approximation error are derived in terms of the components of the a posteriori error estimator. The non-linear projections do not need to be explicitly computed to construct the a posteriori error estimates.
(paper in .pdf format)

6. Ervin, V.J., Lee, H., and Ntasin, L.N., “Analysis of the Oseen-Viscoelastic Fluid Flow Problem,” J. Non-Newtonian Fluid Mech., 127, 157-168, (2005).

Abstract:  In this article we study the numerical approximation of an Oseen type model for  viscoelastic fluid flow. Existence and uniqueness of the continuous and approximate solutions, under a small data assumption, are proved. Error estimates for the numerical approximations are also derived. Numerical experiments are presented which support the error estimates, and which demonstrate the relevance of the small data assumption for the solvability of the continuous and discrete systems.
(paper in .pdf format)

5. Ervin, V.J., and Shepherd, J.J., “Numerical Approximation of the Newtonian Film Blowing Problem,” Computers Math. Applic., 49, 1687-1707, (2005).

Abstract: In this article we study the numerical approximation of a Newtonian model for film blowing. We prove that the approximations for the bubble radius, and the film thickness, converges to the true solution and establish the convergence rates. Numerical results are given which demonstrate the theoretical results obtained.
(paper in .pdf format)

4. Ervin, V.J., and Ntasin, L.N., “A Posteriori Error Estimation and Adaptive Computation of Viscoelastic Fluid Flow,” Numer. Methods for Partial Differential Equations, 21, 297-322, (2005).

Abstract: In this paper, we develop a posteriori error estimates for Finite Element (FE) approximations of viscoelastic fluid flows governed by differential constitutive laws of Giesekus and Oldroyd-B type. We use the general framework developed by Verf\"urth for constructing residual based a posteriori error estimates for nonlinear equations. Numerical experiments using adaptive computations demonstrating the effectiveness of these error estimates are then presented for three examples. The first two examples are problems with known solutions and the third example is, a benchmark problem, the channel flow with a cylindrical obstacle problem. 
(paper in .pdf format)

3. Ervin, V.J., and Miles, W.W., “Approximation of Time-Dependent, Multicomponent, Viscoelastic Fluid Flow,” Comp. Meth. Appl. Mech. Eng., 194, 2229-2255, (2005).

Abstract: In this article we analyse a fully discrete approximation to the time dependent viscoelasticity equations allowing for multicomponent fluid flow.  The Oldroyd B constitutive equation is used to model the viscoelastic stress. For the discretization, time derivatives are replaced by backward difference quotients, and the non-linear terms are linearized by lagging appropriate factors. The modeling equations for the individual fluids are combined into a single system of equations using a continuum surface model.  The numerical approximation is stabilized by using a SUPG approximation for the constitutive equation. Under a small data assumption on the true solution, existence of the approximate solution is proven. A priori error estimates for the approximation in terms of the mesh parameter $h$, the time discretization parameter $\Delta t$, and the SUPG coefficient $\nu$ are also derived. Numerical simulations of viscoelastic fluid flow involving two immiscible fluids are also presented.
(paper in .pdf format)

2. Ervin, V.J., and Miles, W.W., “Approximation of Time-Dependent, Viscoelastic Fluid Flow: SUPG Approximation,” SIAM J. Numer. Anal. 41, 457-486, (2003).

Abstract: In this article we consider the numerical approximation to the time dependent viscoelasticity equations with an Oldroyd B constitutive equation. The approximation is stabilized by using a SUPG approximation for the constitutive equation. We analyse both the semi-discrete and fully discrete numerical approximations. For both discretizations we prove the existence of, and derive a priori error estimates for, the numerical approximations.
(paper in .pdf format)

1. Ervin, V.J., and Heuer, N., “Approximation of Time-Dependent, Viscoelastic Fluid Flow: Crank-Nicolson, Finite Element Approximation,” Numer. Methods for Partial Differential Equations, 20, 248-283, (2003).

Abstract: In this article we analyze a fully discrete approximation to the time dependent viscoelasticity equations with an Oldroyd B constitutive equation in $\real^{\dd}, \, \dd = 2, 3$. We use a Crank--Nicolson discretization for the time derivatives. At each time level a linear system of equations is solved. To resolve the non-linearities we use a three step extrapolation for the prediction of the velocity and stress at the new time level. The approximation is stabilized by using a discontinuous Galerkin approximation for the constitutive equation. For the mesh parameter, $h$, and the
temporal step size, $\Delta t$, sufficiently small and satisfying $\Delta t \le C h^{\dd/4}$, existence of the approximate solution is proven. A priori error estimates for the approximation in terms of $\Delta t$ and $h$ are also derived.
(paper in .pdf format)

General

3. Ervin, V.J., Layton, W.J., and Neda, M., “Numerical Analysis of a Higher Order Time Relaxation Model of Fluids,” to appear International Journal of Numerical Analysis and Modeling.

Abstract: We study the numerical errors in finite element discretizations of a time relaxation model of fluid motion:
u_{t}  +  u \cdot \Grad u  +  \Grad p  -  \nu \Delta u  + \chi u^{*}  = f   ,
and                                                                             Grad \cdot u = 0   .
In this model, introduced by Stolz, Adams and Kleiser, u^{*}  is a generalized fluctuation and $\chi$ the time relaxation parameter. The goal of inclusion of the $\chi u^{*}$ is to drive unresolved fluctuations to zero exponentially.
We study convergence of discretization of the model to the model's solution as $h,\,\delt\,\rightarrow 0$. Next we complement this with an experimental study of the effect the time relaxation term (and a nonlinear extension of it) has on the large scales of a flow near a transitional point. We close by showing that the time relaxation term does not alter shock speeds in the inviscid, compressible case, giving analytical confirmation of a result of Stolz, Adams and Kleiser.
(paper in .pdf format)

2. Ervin, V.J., and Ntasin, L.N., “Improving the Effectivity of Residual Based A Posteriori Error Estimates using a Statistical Approach,” Comp. Meth. Appl. Mech. Eng., 195, 614-631, (2006).

Abstract: For the approximation of differential equations residual based error estimates provide upper bounds (usually gross over estimates) to the true error. In this paper we present a procedure for determining values for the constants in the a posteriori estimates which yield accurate estimates to the true error. Numerical experiments demonstrating the effectiveness of the method are given.
(paper in .pdf format)

1. Brannan, J., Duan, J., Ervin, V.J., and Razoumov, L., “A Weiner-Hopf Approximation Technique for a Multiple Plate Diffraction Problem,” Mathematical Methods in the Applied Sciences, 27, 19-34, (2004).

Abstract: An approximation method is derived for the computation of the acoustic field between a series of parallel plates, subject to a time periodic incident field. The method is based on the Wiener-Hopf method of factorization, with computations involving orthogonal bases of functions that are analytic in the complex half-plane.
(paper in .pdf format)