The purpose of this work is the study of solution techniques for problems involving fractional powers of symmetric coercive elliptic operators in a bounded domain with Dirichlet boundary conditions. These operators can be realized as the Dirichlet to Neumann map for a degenerate/singular elliptic problem posed on a semi-infinite cylinder, which we analyze in the framework of weighted Sobolev spaces. Motivated by the rapid decay of the solution of this problem, we propose a truncation that is suitable for numerical approximation. We discretize this truncation using first degree tensor product finite elements. We derive a priori error estimates in weighted Sobolev spaces. The estimates exhibit optimal regularity but suboptimal order for quasi-uniform meshes. For anisotropic meshes, instead, they are quasi-optimal in both order and regularity. We present numerical experiments to illustrate the method's performance.

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Abstract. We study solution techniques for parabolic equations with fractional diffusion and Caputo fractional time derivative, the latter being discretized and analyzed in a general Hilbert space setting. The spatial fractional diffusion is realized as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on a semi-infinite cylinder in one more spatial dimension. We write our evolution problem as a quasi-stationary elliptic problem with a dynamic boundary condition. We propose and analyze an implicit fully-discrete scheme: first-degree tensor product finite elements in space and an implicit finite difference discretization in time. We prove stability and error estimates for this scheme.

We develop a constructive piecewise polynomial approximation theory in weighted Sobolev spaces with Muckenhoupt weights for any polynomial degree. The main ingredients to derive optimal error estimates for an averaged Taylor polynomial are a suitable weighted Poincaré inequality, a cancellation property and a simple induction argument. We also construct a quasi-interpolation operator, built on local averages over stars, which is well defined for functions in L 1 . We derive optimal error estimates for any polynomial degree on simplicial shape regular meshes. On rectangular meshes, these estimates are valid under the condition that neighboring elements have comparable size, which yields optimal anisotropic error estimates over n-rectangular domains. The interpolation theory extends to cases when the error and function regularity require different weights. We conclude with three applications: nonuniform elliptic boundary value problems, elliptic problems with singular sources, and fractional powers of elliptic operators.

We present three schemes for the numerical approximation of fractional diffusion, which build on different definitions of such a non-local process. The first method is a PDE approach that applies to the spectral definition and exploits the extension to one higher dimension. The second method is the integral formulation and deals with singular non-integrable kernels. The third method is a discretization of the Dunford-Taylor formula. We discuss pros and cons of each method, error estimates, and document their performance with a few numerical experiments.Date: Draft version of July 7, 2017.

We describe and analyze preconditioned steepest descent (PSD) solvers for fourth and sixthorder nonlinear elliptic equations that include p-Laplacian terms on periodic domains in 2 and 3 dimensions. The highest and lowest order terms of the equations are constant-coefficient, positive linear operators, which suggests a natural preconditioning strategy. Such nonlinear elliptic equations often arise from time discretization of parabolic equations that model various biological and physical phenomena, in particular, liquid crystals, thin film epitaxial growth and phase transformations. The analyses of the schemes involve the characterization of the strictly convex energies associated with the equations. We first give a general framework for PSD in generic Hilbert spaces. Based on certain reasonable assumptions of the linear pre-conditioner, a geometric convergence rate is shown for the nonlinear PSD iteration. We then apply the general the theory to the fourth and sixth-order problems of interest, making use of Sobolev embedding and regularity results to confirm the appropriateness of our pre-conditioners for the regularized p-Lapacian problems. Our results include a sharper theoretical convergence result for p-Laplacian systems compared to what may be found in existing works. We demonstrate rigorously how to apply the theory in the finite dimensional setting using finite difference discretization methods. Numerical simulations for some important physical application problems -including thin film epitaxy with slope selection and the square phase field crystal model -are carried out to verify the efficiency of the scheme.

We review the construction and analysis of numerical methods for strongly nonlinear PDEs, with an emphasis on convex and nonconvex fully nonlinear equations and the convergence to viscosity solutions. We begin by describing a fundamental result in this area which states that stable, consistent, and monotone schemes converge as the discretization parameter tends to zero. We review methodologies to construct finite difference, finite element, and semi-Lagrangian schemes that satisfy these criteria, and, in addition, discuss some rather novel tools that have paved the way to derive rates of convergence within this framework.

We design and analyze several Finite Element Methods (FEMs) applied to the Caffarelli-Silvestre extension that localizes the fractional powers of symmetric, coercive, linear elliptic operators in bounded domains with Dirichlet boundary conditions. We consider open, bounded, polytopal but not necessarily convex domains Ω ⊂ R d with d = 1, 2. For the solution to the extension problem, we establish analytic regularity with respect to the extended variable y ∈ (0, ∞). We prove that the solution belongs to countably normed, power-exponentially weighted Bochner spaces of analytic functions with respect to y, taking values in corner-weighted Kondat'ev type Sobolev spaces in Ω. In Ω ⊂ R 2 , we discretize with continuous, piecewise linear, Lagrangian FEM (P 1 -FEM) with mesh refinement near corners, and prove that first order convergence rate is attained for compatible data f ∈ H 1−s (Ω).We also prove that tensorization of a P 1 -FEM in Ω with a suitable hp-FEM in the extended variable achieves log-linear complexity with respect to N Ω , the number of degrees of freedom in the domain Ω. In addition, we propose a novel, sparse tensor product FEM based on a multilevel P 1 -FEM in Ω and on a P 1 -FEM on radical-geometric meshes in the extended variable. We prove that this approach also achieves log-linear complexity with respect to N Ω . Finally, under the stronger assumption that the data is analytic in Ω, and without compatibility at ∂Ω, we establish exponential rates of convergence of hp-FEM for spectral, fractional diffusion operators in energy norm. This is achieved by a combined tensor product hp-FEM for the Caffarelli-Silvestre extension in the truncated cylinder Ω × (0, Y ) with anisotropic geometric meshes that are refined towards ∂Ω. We also report numerical experiments for model problems which confirm the theoretical results. We indicate several extensions and generalizations of the proposed methods to other problem classes and to other boundary conditions on ∂Ω.