Vincent J. Ervin

Numerical Approximation of a Time Dependent, Nonlinear, Space-Fractional Diffusion Equation

In this article we analyze a fully discrete numerical approximation to a time dependent fractional order diffusion equation which contains a nonlocal quadratic nonlinearity. The analysis is performed for a general fractional order diffusion operator. The nonlinear 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.

Coupled Generalized Nonlinear Stokes Flow with Flow through a Porous Medium

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.

A Connection Between Scott-Vogelius and Grad-Div Stabilized Taylor-Hood FE Approximations of the Navier-Stokes Equations

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.

A posteriori error estimators for a two-level finite element method for the Navier-Stokes equations

Two- and multilevel truncated Newton finite element discretizations are presently a very promising approach for approximating the (nonlinear) Navier-Stokes equations describing the equilibrium flow of a viscous, incompressible fluid. Their combination with mesh adaptivity is considered in this article. Specifically, locally calculable a posteriori error estimators are derived, with full mathematical support, for the basic two-level discretization of the Navier-Stokes equations.

Adaptive Defect-Correction Methods for Viscous Incompressible Flow Problems

We consider a defect correction method (DCM) which has been used extensively in applications where solutions have sharp transition regions, such as high Reynolds number fluid flow problems. A reliable a posteriori error estimator is derived for a defect correction method. The estimator is further studied for two examples: (a) the case of a linear-diffusion, nonlinear convection-reaction equation, and (b) the nonlinear Navier--Stokes equations. Numerical experiments are provided which illustrate the utility of the resulting adaptive defect correction method for high Reynolds number, incompressible, viscous flow problems.

On the h-p version of the boundary element method for Symm's integral equation on polygons

Abstract In this paper, we present the numerical implementation of, and numerical experiments for, the Galerkin approximation of Symms integral equation using the h , p and h - p methods. Numerical results obtained using an adaptive algorithm are also given. The theoretical results for these methods are summarized and are compared with the experimental results.

Finite-element modeling of heat transfer in carbon/carbon composites

A finite-element model has been developed to predict the thermal conductivities, parallel and transverse to the fiber axis, of unidirectional carbon/carbon composites. This versatile model incorporates fiber morphology, matrix morphology, fiber/matrix bonding, and random distribution of fibers, porosity, and cracks. The model first examines the effects of the preceding variables on the thermal conductivity at the microscopic level and then utilizes those results to determine the overall thermal conductivity. The model was able accurately to predict the average thermal conductivity of standard pitch-based carbon/carbon composites. The model was also used to study the effect of different composite architectures on the bulk thermal conductivity. The effects of fiber morphology, fiber/matrix interface, and the ratio of transverse fiber conductivity to matrix conductivity on the overall composite conductivity was examined.

Approximation of Time-Dependent Viscoelastic Fluid Flow: SUPG Approximation

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 streamline upwind Petrov--Galerkin (SUPG) approximation for the constitutive equation. We analyze both the semidiscrete and fully discrete numerical approximations. For both discretizations we prove the existence of, and derive a priori error estimates for, the numerical approximations.

Collocation with Chebyshev polynomials for a hypersingular integral equation on an interval

Abstract A collocation method for a first-kind integral equation with a hypersingular kernel on an interval is analysed. Chebyshev polynomials of the second kind are used as the basis functions for the approximation, and the collocation points are chosen to be Chebyshev quadrature points. In our analysis we introduce Sobolev norms that reflect the singular structure of the exact solution at the endpoints of the interval. Numerical experiments are presented which underline the theoretical estimates.

On the stochastic Kuramoto—Sivanshinsky equation

In this article we study the solution of the Kuramoto–Sivashinsky equation on a bounded interval subject to a random forcing term. We show that a unique solution to the equation exists for all time and depends continuously on the initial data.