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.

In this article, we discuss the steady state fractional advection dispersion equation (FADE) on bounded domains in R d . Fractional differential and integral operators are defined and analyzed. Appropriate fractional derivative spaces are defined and shown to be equivalent to the fractional dimensional Sobolev spaces. A theoretical framework for the variational solution of the steady state FADE is presented. Existence and uniqueness results are proven, and error estimates obtained for the finite element approximation.

The problem of providing succinct approximate descriptions of given bounded subsets of RX can be solved by application of the contraction mapping principle.We present an application of the contraction mapping principle which yields succinct descriptions, approximations, and reconstructions for complicated sets such as fractals (1)

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.

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.

In this report we investigate the regularity of the solution to the fractional diffusion, advection, reaction equation on a bounded domain in R 1 . The analysis is performed in the weighted Sobolev spaces, H s (a,b) (I). Three different characterizations of H s (a,b) (I) 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 α 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.In this article we present the general regularity results for (1.3),(1.4), in appropriately weighted Sobolev spaces. The analysis establishes that the presence of a reaction term (i.e. c(x) = 0) limits the regularity of the solution, regardless of the smoothness of the right hand side function, f (x). This reduction in regularity is greater (by a factor of 2) when an advective term (i.e. b(x) = 0) appears in (1.3). This behavior of the solution is in sharp contrast to that for the integer order (α = 2) diffusion, advection, reaction equation. In that case, assuming b(x) and c(x) are sufficiently regular, for the right hand side function f ∈ H s (I) the solution lies in H s+2 (I).The results we present herein extend those in [13] for the fractional diffusion equation, and those in [22] for the fractional Laplacian equation with a constant advection and reaction term. The proofs given are significantly different that those used in [21,22].The analysis of (1.3),(1.4) is most appropriately performed in weighted Sobolev spaces (due to the singular behavior of the solution at the endpoints). There are different ways to define the weighted Sobolev spaces: (i) using interpolation (Section 3), (ii) using an appropriate basis (Section 4), (iii)

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.

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.1. Introduction. Simulations in critical settings often struggle with numerical artifacts created by under resolution of the spacial scales and temporal dynamics in the model, e.g., Brown and Minon [BM95]. Often, as soon as there are sufficient computational resources for full resolution, the demands of the application force coupling to other processes, making the target simulation again under resolved.