Ambartsumyan, I., Ervin, V.J., Nguyen, T., and Yotov, I., "A nonlinear Stokes-Biot model for the interaction of a non-Newtonian fluid with poroelastic media," ESIAM: Mathematical Modeling and Numerical Analysis, 53 (6), 1915-1955, (2019).
Abstract: We develop and analyze a model for the interaction of a quasi-Newtonian free fluid with a poroelastic medium. The flow in the fluid region is described by the nonlinear Stokes equations and in the poroelastic medium by the nonlinear quasi-static Biot model. Equilibrium and kinematic conditions are imposed on the interface. We establish existence and uniqueness of a solution to the weak formulation and its semidiscrete continuous-in-time finite element approximation. We present error analysis, complemented by numerical experiments. (paper in .pdf format)
Ervin, V.J., Kubacki, M., Layton, W., Moraiti, M., Si, Z., and Trenchea, C., "Partitioned penalty methods for the evolutionary Stokes-Darcy-transport problem," Numerical Methods for Partial Differential Equations, 35, 349-374, (2019).
Abstract: There has been a surge of work on models for coupling surface-water with groundwater flows which is at its core the Stokes-Darcy problem, as well as methods for uncoupling the problem into subdomain, subphysics solves. The resulting (Stokes-Darcy) fluid velocity is important because the flow transports contaminants. The numerical analysis and algorithm development for the evolutionary transport problem has, however, focused on a quasi-static Stokes-Darcy model and a single domain (fully coupled) formulation of the transport equation. This report presents a numerical analysis of a partitioned method for contaminant transport for the fully evolutionary system. The algorithm studied is unconditionally stable with one subdomain solve per step. Numerical experiments are given using the proposed algorithm, that investigate the effects of the penalty parameters on the convergence of the approximations. (paper in .pdf format)
Ervin, V.J., and Ruiz-Ramirez, J., "A deposition model coupling Stokes’ and Darcy’s equations with nonlinear deposition," Journal of Computational and Applied Mathematics, 340, 151-172, (2018).
Abstract: In this work we investigate a filtration process whereby particulate is deposited in the flow domain, causing the porosity of the region to decrease. The fluid flow is modeled as a coupled Stokes-Darcy flow problem and the deposition (in the Darcy domain) is modeled using a nonlinear equation for the porosity. Existence and uniqueness of a solution to the governing equations is established. Additionally, the nonnegativity and boundedness of the porosity is shown. A finite element approximation scheme that preserves the nonnegativity and boundedness of the porosity is investigated. Accompanying numerical experiments support the analytical findings. (paper in .pdf format)
Ervin, V.J., Kubacki, M., Layton W., Moraiti, M., Si, Z., and Trenchea, C., "On limiting behavior of contaminant transport models in coupled surface and groundwater flows," Axioms, 4, 518-529, (2015).
Abstract: There has been a surge of work on models for coupling surface-water with groundwater flows which is at its core the Stokes-Darcy problem. The resulting (Stokes-Darcy) fluid velocity is important because the flow transports contaminants. The~analysis of models including the transport of contaminants has, however, focused on a quasi-static Stokes-Darcy model. Herein we consider the fully evolutionary system including contaminant transport and analyze its quasi-static limits. (paper in .pdf format)
Crowder, T.R., and Ervin, V.J., "Numerical simulations of fluid pressure in the human eye," Appl. Math. Comput., 219, 11119-11133, (2013).
Abstract: In this article we present computational results for the pressure in the human eye. Pressure computations for different flow rates of the aqueous humor, viscosity of the aqueous humor, and permeability of the trabecular meshwork are given. The fluid flow is assumed to be axisymmetric, and modeled as a coupled systems of Stokes and Darcy fluid flow equations, represented the fluid flow in the anterior cavity and trabecular meshwork, respectively. Rewriting the problem in cylindrical coordinates reduces the 3-D problem to a problem in 2-D. Computations are also given for varying angles between the base of the iris and the trabecular meshwork. (paper in .pdf format)
Ervin, V.J., "Approximation of coupled Stokes-Darcy flow in an axisymmetric domain," Comp. Meth. Appl. Mech. Eng., 258, 96-108, (2013).
Abstract: In this article we investigate the numerical approximation of coupled Stokes and Darcy fluid flow equations in an axisymmetric domain. The fluid flow is assumed to be axisymmetric. Rewriting the problem in cylindrical coordinates reduces the 3-D problem to a problem in 2-D. This reduction to 2-D requires the numerical analysis to be studied in suitably weighted Hilbert spaces. In this setting we show that the proposed approximation scheme has a unique solution, and derive corresponding a priori error estimate. Computations for an example with a know solution are presented which support the a priori error estimate. Computations are also given for a model of fluid flow in the eye. (paper in .pdf format)
Ervin, V.J., Jenkins, E.W., and Lee, H., "Approximation of the Stokes-Darcy system by optimization," J. Sci. Comput., 59, 775-794, (2014).
Abstract: A solution algorithm for the linear/nonlinear Stokes-Darcy coupled problem is proposed and investigated. The coupled system is formulated as a constrained optimal control problem, where a flow balance is forced across the interface, inflow, and outflow boundaries by minimizing a suitably defined functional. Optimization is achieved by exploiting a Neumann type boundary condition imposed on each subproblem as a control. A numerical algorithm is presented for a least squares functional whose solution yields a minimizer of the constrained optimization problem. Numerical experiments are provided to validate accuracy and efficiency of the algorithm. (paper in .pdf format)
Ervin, V.J., Jenkins, E.W., and Sun, S., "Coupling Non-linear Stokes and Darcy Flow using Mortar Finite Elements," Appl. Numer. Math., 61, 1198-1222, (2011).
Abstract: We study a system composed of a non-linear Stokes flow in one subdomain coupled with a non-linear porous medium flow in another subdomain. Special attention is paid to the mathematical consequence of the shear-dependent fluid viscosity for the Stokes flow and the velocity-dependent effective viscosity for the Darcy flow. Motivated by the physical setting, we consider the case where only flow rates are specified on the inflow and outflow boundaries in both subdomains. We recast the coupled Stokes-Darcy system as a reduced matching problem on the interface using a mortar space approach. We prove a number of properties of the nonlinear interface operator associated with the reduced problem, which directly yield the existence, uniqueness and regularity of a variational solution to the system. We further propose and analyze a numerical algorithm based on mortar finite elements for the interface problem and conforming finite elements for the subdomain problems. Optimal a priori error estimates are established for the interface and subdomain problems, and a number of compatibility conditions for the finite element spaces used are discussed. Numerical simulations are presented to illustrate the flexibility of the proposed algorithm and to compare two treatments of the defective boundary conditions. (paper in .pdf format)
Ervin, V.J., Jenkins, E.W., and Sun, S., "Coupled Generalized Non-linear Stokes Flow with flow through a Porous Media," SIAM J. Numer. Anal., 47, 929-952, (2009).
Abstract: In this article, we analyze the flow of a fluid through a coupled Stokes-Darcy domain. The fluid in each domain is non-Newtonian, modeled by the generalized nonlinear Stokes equation in the free flow region and the generalized nonlinear Darcy equation in the porous medium. A flow rate is specified along the inflow portion of the free flow boundary. We show existence and uniqueness of a variational solution to the problem. We propose and analyze an approximation algorithm and establish a-priori error estimates for the approximation. (paper in .pdf format)
Ervin, V.J., Lee, H., and Ruiz-Ramirez, J., "Nonlinear Darcy fluid flow with deposition," Journal of Computational and Applied Mathematics, 309, 79-94, (2017).
Abstract: In this article we consider a model of a filtration process. The process is modeled using the nonlinear Darcy fluid flow equations with a varying permeability, coupled with a deposition equation. Existence and uniqueness of the solution to the modeling equations is established. A numerical approximation scheme is proposed and analyzed, with an a priori error estimate derived. Numerical experiments are presented which support the obtained theoretical results. (paper in .pdf format)
Ervin, V.J., Lee, H., and Salgado, A.J., "Generalized Newtonian fluid flow through a porous medium," J. Math. Anal. Appl., 433, 603-621, (2016).
Abstract: We present a model for generalized Newtonian fluid flow through a porous medium. In the model the dependence of the fluid viscosity on the velocity is replaced by a dependence on a smoothed (locally averaged) velocity. With appropriate assumptions on the smoothed velocity, existence of a solution to the model is shown. Two examples of smoothing operators are presented in the appendix. A numerical approximation scheme is presented and an a priori error estimate derived. A numerical example is given illustrating the approximation scheme and the a priori error estimate. (paper in .pdf format)
Chrispell, J.C, Ervin, V.J., and Jenkins, E.W., "A Fractional Step Θ-method for Viscoelastic Fluid Flow using a SUPG approximation," International Journal of Computational Science, 2, 336-351, (2008).
Abstract: In this article a fractional step Θ-method is described and studied for the approximation of time dependent viscoelastic fluid flow equations, using the Johnson-Segalman constitutive model. The Θ-method implementation allows the velocity and pressure approximations to be decoupled from the stress, reducing the number of unknowns resolved at each step of the method. The constitutive equation is stabilized using a Streamline Upwinded Petrov-Galerkin (SUPG)-method. A priori error estimates are given for the approximation scheme. Numerical computations supporting the theoretical results and demonstrating the $\theta$-method are also presented. (paper in .pdf format)
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," Comp. Meth. Appl. Mech. Eng., 197, 2886-2900, (2008).
Abstract: In this work a dual-mixed approximation of a nonlinear generalized Stokes problem is studied. The problem is analyzed in Sobolev spaces which arise naturally in the problem formulation. Existence and uniqueness results are given and error estimates are derived. It is shown that both lowest-order and higher-order mixed finite elements are suitable for the approximation method. Numerical experiments that support the theoretical results are presented. (paper in .pdf format)
Ervin, V.J., Howell, J.S. and Lee, H., "A Two-Parameter Defect-Correction Method for Computation of Steady-State Viscoelastic Fluid Flow," Appl. Math. Comput., 196, 818-834, (2008).
Abstract: The numerical simulation of viscoelastic fluid flow becomes more difficult as a physical parameter, the Weissenberg number, increases. Specifically, at a Weissenberg number larger than a critical value, the iterative nonlinear solver fails to converge. In this paper a two-parameter defect-correction method for viscoelastic fluid flow is presented and analyzed. In the defect step the Weissenberg number is artifically reduced to solve a stable nonlinear problem. The approximation is then improved in the correction step using a linearized correction iteration. Numerical experiments support the theoretical results and demonstrate the effectiveness of the method. (paper in .pdf format)
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).
Abstract: In this article, we analyze the fractional-step Θ-method for the time-dependent convection-diffusion equation. In our implementation, we completely separate the convection operator from the diffusion operator, and we stabilize the convective solve using a streamline upwinded Petrov-Galerkin (SUPG) method. We establish a-priori error estimates and show that optimal values of Θ yield a scheme that is second order in time. Numerical results are presented which demonstrate the method and support the theoretical results. (paper in .pdf format)
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).
Abstract: In this article we study the numerical approximation of a quasi-Newtonian Stokes flow problem where only the flow rates are specified at the inflow and outflow boundaries. A variational formulation of the problem, using Lagrange multipliers to enforce the stated flow rates, is given. Existence and uniqueness of the solution to the continuous, and discrete, variational formulations is shown. An error analysis for the numerical approximation is also given. Numerical computations are included which demonstrate the approximation scheme studied. (paper in .pdf format)
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)
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)
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)
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)
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)
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)
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)
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 R^{\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)
Case, M.A, Ervin, V.J., Linke, A., and Rebholz, L.G., "A Connecton Between Scott-Vogelius and Grad-Div Stabilized Taylor-Hood Approximations of the Navier-Stokes Equations," SIAM J. Numer. Anal., 49, 1461-1481 (2011).
Abstract: This article studies two methods for obtaining excellent mass conservation in finite element computations of the Navier-Stokes equations using continuous velocity fields. With a particular mesh construction, the Scott-Vogelius element pair has recently been shown to be inf-sup stable and have optimal approximation properties, while also providing pointwise mass conservation. We present herein the first numerical tests of this element pair for the time dependent Navier-Stokes equations. We also prove that the limit of the grad-div stabilized Taylor-Hood solutions to the Navier-Stokes problem converges to the Scott-Vogelius solution as the stabilization parameter tends to infinity. That is, we provide theoretical justification that choosing the grad-div parameter large does not destroy the solution. Numerical tests are provided which verify the theory, and show how both Scott-Vogelius and grad-div stabilized Taylor-Hood (with large stabilization parameter) elements can provide accurate results with excellent mass conservation for Navier-Stokes approximations. (paper in .pdf format)
Ervin, V.J., Layton, W.J., and Neda, M., "Numerical Analysis of Filter Based Stabilization for Evolution Equations," SIAM J. Numer. Anal., 50, 2307-2335, (2012).
Abstract: We consider filter based stabilization for evolution equations (in general) and for the Navier-Stokes equations (in particular). The first method we consider is to advance in time one time step by a given method and then to apply an (uncoupled and modular) filter to get the approximation at the new time level. This filter based stabilization, although algorithmically appealing, is viewed in the literature as introducing far too much numerical dissipation to achieve a quality approximate solution. We show that this is indeed the case. We then consider a modification: Evolve one time step, Filter, Deconvolve then Relax to get the approximation at the new time step. We give a precise analysis of the numerical diffusion and error in this process and show it has great promise, confirmed in several numerical experiments. (paper in .pdf format)
Case, M.A, Ervin, V.J., Linke, A., Rebholz, L.G., and Wilson, N.E., "Stable Computing with an Enhanced Physics Based Scheme for the 3d Navier-Stokes Equations," International Journal of Numerical Analysis and Modeling, 8, 118-136, (2011).
Abstract: We study extensions of the energy and helicity preserving scheme for the 3D Navier-Stokes equations, developed in \cite{Reb07b}, to a more general class of problems. The scheme is studied together with stabilizations of grad-div type in order to mitigate the effect of the Bernoulli pressure error on the velocity error. We prove stability, convergence, discuss conservation properties, and present numerical experiments that demonstrate the advantages of the scheme. (paper in .pdf format)
Ervin, V.J., Layton, W.J., and Neda, M., "Numerical Analysis of a Higher Order Time Relaxation Model of Fluids," International Journal of Numerical Analysis and Modeling, 4, 648-670, (2007).
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)
Zheng, X., Ervin, V.J., and Wang, H., "Optimal Petrov-Galerkin spectral approximation method for the fractional diffusion, advection, reaction equation on a bounded interval," J. Sci. Comput., 86, 29, https://doi.org/10.1007/s10915-020-01366-y, (2021).
Abstract: In this paper we investigate the numerical approximation of the fractional diffusion, advection, reaction equation on a bounded interval. Recently the explicit form of the solution to this equation was obtained. Using the explicit form of the boundary behavior of the solution and Jacobi polynomials, a Petrov-Galerkin approximation scheme is proposed and analyzed. Numerical experiments are presented which support the theoretical results, and demonstrate the accuracy and optimal convergence of the approximation method. (paper in .pdf format)
Ervin, V.J., "Regularity of the solution to fractional diffusion, advection, reaction equations in weighted Sobolev," J. Differential Equations, 278, 294-325, https://doi.org/10.1016/j.jde.2020.12.034, (2021).
Abstract: In this article we investigate the regularity of the solution to the fractional diffusion, advection, reaction equation on a bounded domain in $\mathbb{R}^{1}$. The analysis is performed in the weighted Sobolev spaces, $H_{(a , b)}^{s}(\mrI)$. Three different characterizations of $H_{(a , b)}^{s}(\mrI)$ are presented, together with needed embedding theorems for these spaces. The analysis shows that the regularity of the solution is bounded by the endpoint behavior of the solution, which is determined by the parameters $\alpha$ and $r$ defining the fractional diffusion operator. Additionally, the analysis shows that for a sufficiently smooth right hand side function, the regularity of the solution to fractional diffusion reaction equation is lower than that of the fractional diffusion equation. Also, the regularity of the solution to fractional diffusion advection reaction equation is two orders lower than that of the fractional diffusion reaction equation. (paper in .pdf format)
Yang, S., Chen, H., Ervin, V.J., and Wang, H., "Solvability and approximation of two-sided conservative fractional diffusion problems with variable coefficient based on least squares," to appear Applied Mathematics and Computation, (2021).
Abstract: In this article, we investigate the solvability theory and numerical simulation for two-side conservative fractional diffusion equations (CFDE) with variable-coefficient $K(x)$. We introduce $u = -K D p$ as an intermediate variable to isolate $K(x)$ from the nonlocal operator, and then apply the least-squares to formulate a mixed-type variational formulation for the unknown and the intermediate variable. Correspondingly, the admissible space for the solution is decomposed as a direct sum of a regular fractional Sobolev space and a space spanned by two known kernel-dependent singular functions by proving that the two-side fractional derivative operator is a bijective mapping from its admissible space to $L^{2}(\Omega)$. The solution $p$ and $u$ then are represented as a sum of a regular part and a kernel-dependent singular part with two undetermined constant coefficients, which can be expressed by prescribed boundary conditions and derived orthogonal decomposition for $L^{2}(\Omega)$. respectively. Thus, a new regularity theory for the solution is established in terms of the right-hand side only, which extends those regularity results of one side CFDE in [17, 39], and of fractional Laplace operator corresponding to $\theta = 1/2$ in [1, 14] to more general CFDE with variable diffusive coefficients for $0 < \theta < 1$. Then, we design a kernel-independent least-squares mixed finite element procedure (LSMFE). Theoretical analysis and numerical experiments conducted in this article demonstrate that the LSMFE can capture the singular part of the solution exactly, approximate the solution with optimal-order accuracy, and be easily implemented. (paper in .pdf format)
Zheng, X., Ervin, V.J., and Wang, H., "An indirect finite element method for variable-coefficient space-fractional diffusion equations and its optimal order error estimates," Communications on Applied Mathematics and Computation, 2 (1), 147-162, DOI.org/10.1007/s42967-019-00037-6, (2020).
Abstract: We study an indirect finite element approximation for two-sided space-fractional diffusion equations in one space dimension. By the representation formula of the solutions $u(x)$ to the proposed variable coefficient models in terms of $v(x)$, the solutions to the constant coefficient analogues, we apply finite element methods for the constant coefficient fractional diffusion equations to solve for the approximations $v_h(x)$ to $v(x)$ and then obtain the approximations $u_h(x)$ of $u(x)$ by plugging $v_h(x)$ into the representation of $u(x)$. Optimal-order convergence estimates of $u(x)-u_h(x)$ are proved in both $L^2$ and $H^{\alpha/2}$ norms. Several numerical experiments are presented to demonstrate the sharpness of the derived error estimates. (paper in .pdf format)
Zheng, X., Ervin, V.J., and Wang, H., "Numerical approximations for the variable coefficient fractional diffusion equations with non-smooth data," Computational Methods in Applied Mathematics, 20(3), 573-589, https://doi.org/10.1515/cmam-2019-0038, (2020).
Abstract: In this article we study the numerical approximation of a variable coefficient fractional diffusion equation. Using a change of variable, the variable coefficient fractional diffusion equation is transformed into a constant coefficient fractional diffusion equation of the same order. The transformed equation retains the desirable stability property of being an elliptic equation. A spectral approximation scheme is proposed and analyzed for the transformed equation, with error estimates for the approximated solution derived. An approximation to the unknown of the variable coefficient fractional diffusion equation is then obtained by post processing the computed approximation to the transformed equation. Error estimates are also presented for the approximation to the unknown of the variable coefficient equation with both smooth and non-smooth diffusivity coefficient and right-hand side. Three numerical experiments are given whose convergence results are in strong agreement with the theoretically derived estimates. (paper in .pdf format)
Zheng, X., Ervin, V.J., and Wang, H., "Wellposedness of the two-sided variable coefficient Caputo flux fractional diffusion equation and error estimate of its spectral approximation," Appl. Numer. Math., 153, 234-247, (2020).
Abstract: In this article a two-sided variable coefficient fractional diffusion equation (FDE) is investigated, where the variable coefficient occurs outside of the fractional integral operator. Under a suitable transformation the variable coefficient equation is transformed to a constant coefficient equation. Then, using the spectral decomposition approach with Jacobi polynomials, we proved the wellposedness of the model and the regularity of its solution. A spectral approximation scheme is proposed and the accuracy of its approximation studied. Two numerical experiments are presented to demonstrate the derived error estimates. (paper in .pdf format)
Jia, L., Chen, H., and Ervin, V.J., "Existence and Regularity of solutions to 1-D Fractional Order Diffusion Equations," Electronic J. Differential Equations, 93, 1-21, (2019).
Abstract: In this article we investigate the existence and regularity of 1-d steady state fractional order diffusion equations. Two models are investigated: the Riemann-Liouville fractional diffusion equation, and the Riemann-Liouville-Caputo fractional diffusion equation. For these models we explicitly show how the regularity of the solution depends upon the right hand side (rhs) function. We also establish for which Dirichlet and Neumann boundary conditions the models are well posed. (paper in .pdf format)
Zheng, X., Ervin, V.J., and Wang, H., "Spectral approximation of a variable coefficient fractional diffusion equation in one space dimension," Appl. Math. Comput., 361, 98-111, (2019).
Abstract: In this article we consider the approximation of a variable coefficient (two-sided) fractional diffusion equation (FDE), having unknown $u$. By introducing an intermediate unknown, $q$, the variable coefficient FDE is rewritten as a lower order, constant coefficient FDE. A spectral approximation scheme, using Jacobi polynomials, is presented for the approximation of $q$, $q_{N}$. The approximate solution to $u$, $u_{N}$, is obtained by post processing $q_{N}$. An a priori error analysis is given for $(q \, - \, q_{N})$ and $(u \, - \, u_{N})$. Two numerical experiments are presented whose results demonstrate the sharpness of the derived error estimates. (paper in .pdf format)
Ervin, V.J., Heuer, N. and Roop, J.P., "Regularity of the Solution to 1-D Fractional Order Diffusion Equations," Mathematics of Computation, 87, 2273-2294, (2018).
Abstract: In this article we investigate the solution of the steady-state fractional diffusion equation on a bounded domain in $\real^{1}$. From an analysis of the underlying model problem, we postulate that the fractional diffusion operator in the modeling equations is neither the Riemann-Liouville nor the Caputo fractional differential operators. We then find a closed form expression for the kernel of the fractional diffusion operator which, in most cases, determines the regularity of the solution. Next we establish that the Jacobi polynomials are pseudo eigenfunctions for the fractional diffusion operator. A spectral type approximation method for the solution of the steady-state fractional diffusion equation is then proposed and studied. (paper in .pdf format)
Ervin, V.J., Fuhrer, T., Heuer, N., and Karkulik, M., "DPG method with optimal test functions for a fractional advection diffusion equation," J. Sci. Comput., 72, 568-585, (2017).
Abstract: We develop an ultra-weak variational formulation of a fractional advection diffusion problem in one space dimension and prove its well-posedness. Based on this formulation, we define a DPG approximation with optimal test functions and show its quasi-optimal convergence. Numerical experiments confirm expected convergence properties, for uniform and adaptively refined meshes. (paper in .pdf format)
Ervin, V.J., Heuer, N. and Roop, J.P., "Numerical Approximation of a Time Dependent, Non-linear, Fractional Order Diffusion Equation," 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)
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)
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.Bentley, A., and Ervin, V.J., "Approximation of the Axisymmetric Elasticity Equations," Comput. Methods. Appl. Mech. Engrg., 374, 113581, 24pp., https://doi.org/10.1016/j.cma.2020.113581, (2021).
Abstract: In this article we consider the linear elasticity problem in an axisymmetric three dimensional domain, with data which are axisymmetric and have zero angular component. The weak formulation of the three dimensional problem reduces to a two dimensional problem on the meridian domain, involving weighted integrals. The problem is formulated in a mixed method framework with both the stress and displacement treated as unknowns. The symmetry condition for the stress tensor is weakly imposed. Well posedness of the continuous weak formulation and its discretization are shown. Two approximation spaces are discussed and corresponding numerical computations are presented. (paper in .pdf format)
Ervin, V.J., Macias-Diaz, J.E., and Ruiz-Ramirez, J., "A positive and bounded finite element approximation of the generalized Burgers-Huxley equation," J. Math. Anal. Appl., 424, 1143-1160, (2014).
Abstract: We present a finite element scheme capable of preserving the nonnegative and bounded solutions of the generalized Burgers-Huxley equation. Proofs of existence and uniqueness of a solution to the continuous problem together with some results concerning the boundedness and the nonnegativity of the solution are given. Under appropriate conditions on the mesh and the initial and boundary data, boundedness and nonnegativity of the finite element approximation are established. An a priori error estimate for the approximation is also derived. Numerical experiments are presented which support the derived theoretical results. (paper in .pdf format)
Ervin, V.J., "Approximation of Axisymmetric Darcy Flow," SIAM J. Numer. Anal., 51, 1421-1442, (2013).
Abstract: In this article we investigate the numerical approximation of the Darcy equations in an axisymmetric domain, subject to axisymmetric data. Rewriting the problem in cylindrical coordinates reduces the 3-D problem to a problem in 2-D. This reduction to 2-D requires the numerical analysis to be studied in suitably weighted Hilbert spaces. In this setting the Raviart-Thomas (RT) and Brezzi-Douglas-Marini (BDM) approximation pairs are shown to be LBB stable, and corresponding a priori error estimates derived. Presented numerical experiments confirm the predicted rates of convergence for the RT and BDM approximations. (paper in .pdf format)
Ervin, V.J., and Jenkins, E.W., "Stenberg's sufficiency criteria for the LBB condition for Axisymmetric Stokes Flow," JMAA, 398, 421-437, (2013).
Abstract: In this article we investigate the LBB condition for axisymmetric flow problems. Specifically, the sufficiency condition for approximating pairs to satisfy the LBB condition established by Stenberg in the Cartesian coordinate setting is presented for the cylindrical coordinate setting. For the cylindrical coordinate setting, the Taylor-Hood ($k = 2$) and conforming Crouzeix-Raviart elements are shown to be LBB stable. A priori error bounds for approximations to the axisymmetric Stokes flow problem using Taylor-Hood and Crouzeix-Raviart elements are given. The computed numerical convergence rates for the error for an axisymmetric Stokes flow problem support the theoretical results. (paper in .pdf format)
Ervin, V.J., "Computational Bases for RT_{k} and BDM_{k} on Triangles," Computers Math. Applic., 64, 2765-2774, (2012).
Abstract: In this article, we derive computational bases for Raviart-Thomas ($RT$) and Brezzi-Douglas-Marini ($BDM$) (vector) approximation spaces on a triangulation of a domain in R^{2}. The basis functions, defined on the reference triangle, have a Lagrangian property. The continuity of the normal component of the approximation across the edges in the triangulation is satisfied by the use of the Piola transformation and the Lagrangian property of the basis functions. A numerical example is given demonstrating the approximation property of the bases. (paper in .pdf format)
Ervin, V.J., and Jenkins, E.W., "Stabilized Approximation to Time Dependent Conservation Equations via Filtering," Appl. Math. Comput., 217, 7282-7294, (2011).
Abstract: We analyze a stabilization technique for advection dominated flow problems. Of particular interest are coupled parabolic/hyperbolic problems, when the diffusion coefficient is zero in part of the domain. The unstabilized, computed approximations of these problems are highly oscillatory, and several techniques have been proposed and analyzed to mitigate the effects of the subgrid errors that contribute to the oscillatory behavior. In this paper, we modify a time-relaxation algorithm proposed in \cite{ada021} and further studied in \cite{erv071}. Our modification introduces the relaxation operator as a post-processing step. The operator is not time-dependent, so the discrete (relaxation) system need only be factored once. We provide convergence analysis for our algorithm along with numerical results for several model problems. (paper in .pdf format)
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)
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)