Finite element approximation of high-dimensional transport-dominated diffusion problems

High-dimensional partial differential equations with nonnegative characteristic form arise in numerous mathematical models in science. In problems of this kind, the computational challenge of beating the exponential growth of complexity as a function of dimension is exacerbated by the fact that the problem may be transport-dominated. We develop the analysis of stabilised sparse finite element methods for such high-dimensional, non-self-adjoint and possibly degenerate partial differential equations.\ud \ud (Presented as an invited lecture under the title "Computational multiscale modelling: Fokker-Planck equations and their numerical analysis" at the Foundations of Computational Mathematics conference in Santander, Spain, 30 June - 9 July, 2005.

A Gagliardo-Nirenberg inequality, with application to duality-based a posteriori error estimation in the L1 norm

We establish the Gagliardo-Nirenberg-type multiplicative interpolation inequality \[ \|v\|_{{\rm L}1(\mathbb{R}^n)} \leq C \|v\|^{1/2}_{{\rm Lip}'(\mathbb{R}^n)} \|v\|^{1/2}_{{\rm BV}(\mathbb{R}^n)}\qquad \forall v \in {\rm BV}(\mathbb{R}^n), \] where $C$ is a positive constant, independent of $v$. We then use a local version of this inequality to derive an a posteriori error bound in the ${\rm L}1(\Omega')$ norm, with $\bar\Omega' \subset\Omega=(0,1)^n$, for a finite-element approximation to a boundary value problem for a first-order linear hyperbolic equation, under the limited regularity requirement that the solution to the problem belongs to ${\rm BV}(\Omega)$.\ud \ud Dedicated to Professor Boško S Jovanovic on the occasion of his sixtieth birthda

Finite element approximation of steady flows of generalized Newtonian fluids with concentration-dependent power-law index

We consider a system of nonlinear partial differential equations describing the motion of an incompressible chemically reacting generalized Newtonian fluid in three space dimensions. The governing system consists of a steady convection-diffusion equation for the concentration and a generalized steady power-law-type fluid flow model for the velocity and the pressure, where the viscosity depends on both the shear-rate and the concentration through a concentration-dependent power-law index. The aim of the paper is to perform a mathematical analysis of a finite element approximation of this model. We formulate a regularization of the model by introducing an additional term in the conservation-of-momentum equation and construct a finite element approximation of the regularized system. We show the convergence of the finite element method to a weak solution of the regularized model and prove that weak solutions of the regularized problem converge to a weak solution of the original problem.Comment: arXiv admin note: text overlap with arXiv:1703.0476

hp-version interior penalty DGFEMs for the biharmonic equation

We construct hp-version interior penalty discontinuous Galerkin finite element methods (DGFEMs) for the biharmonic equation, including symmetric and nonsymmetric interior penalty discontinuous Galerkin methods and their combinations: semisymmetric methods. Our main concern is to establish the stability and to develop the a priori error analysis of these methods. We establish error bounds that are optimal in h and slightly suboptimal in p. The theoretical results are confirmed by numerical experiments

One-parameter discontinuous Galerkin finite element discretisation of quasilinear parabolic problems

We consider the analysis of a one-parameter family of $hp$--version discontinuous Galerkin finite element methods for the numerical solution of quasilinear parabolic equations of the form u'-\na\cdot\set{a(x,t,\abs{\na u})\na u}=f(x,t,u) on a bounded open set \om\in\re^d, subject to mixed Dirichlet and Neumann boundary conditions on \pr\om. It is assumed that $a$ is a real--valued function which is Lipschitz-continuous and uniformly monotonic in its last argument, and $f$ is a real-valued function which is locally Lipschitz-continuous and satisfies a suitable growth condition in its last argument; both functions are measurable in the first and second arguments. For quasi--uniform $hp$--meshes, if u\in \H^1(0,T;\H^k(\om))\cap\L^\infty(0,T;\H^1(\om)) with $k\geq 3\frac{1}{2}$, for discontinuous piecewise polynomials of degree not less than 1, the approximation error, measured in the broken $H^1$ norm, is proved to be the same as in the linear case: $\mathscr{O}(h^{s-1}/p^{k-3/2})$ with $1\leq s\leq\min\set{p+1,k}$

Adaptive Galerkin approximation algorithms for partial differential equations in infinite dimensions

Space-time variational formulations of infinite-dimensional Fokker-Planck (FP) and Ornstein-Uhlenbeck (OU) equations for functions on a separable Hilbert space $H$ are developed. The well-posedness of these equations in the Hilbert space $L^{2}(H,\mu)$ of functions on $H$, which are square-integrable with respect to a Gaussian measure $\mu$ on $H$, is proved. Specifically, for the infinite-dimensional FP equation, adaptive space-time Galerkin discretizations, based on a tensorized Riesz basis, built from biorthogonal piecewise polynomial wavelet bases in time and the Hermite polynomial chaos in the Wiener-Itô decomposition of $L^{2}(H,\mu)$, are introduced and are shown to converge quasioptimally with respect to the nonlinear, best $N$-term approximation benchmark. As a consequence, the proposed adaptive Galerkin solution algorithms perform quasioptimally with respect to the best $N$-term approximation in the finite-dimensional case, in particular. All constants in our error and complexity bounds are shown to be independent of the number of "active" coordinates identified by the proposed adaptive Galerkin approximation algorithms

Discontinuous Galerkin finite element approximation of Hamilton-Jacobi-Bellman equations with Cordès coefficients

We propose an hp-version discontinuous Galerkin finite element method for fully nonlinear second-order elliptic Hamilton-Jacobi-Bellman equations with Cord�ès coefficients. The method is proven to be consistent and stable, with convergence rates that are optimal with respect to mesh size, and suboptimal in the polynomial degree by only half an order. Numerical experiments on problems with strongly anisotropic diffusion coefficients illustrate the accuracy and computational efficiency of the scheme. An existence and uniqueness result for strong solutions of the fully nonlinear problem, and a semismoothness result for the nonlinear operator are also provided

hp-Version discontinuous Galerkin finite element methods for semilinear parabolic problems

We consider the hp-version interior penalty discontinuous Galerkin finite element method (hp-DGFEM) for semilinear parabolic equations with mixed Dirichlet and Neumann boundary conditions. Our main concern is the error analysis of the hp--DGFEM on shape--regular spatial meshes. We derive error bounds under various hypotheses on the regularity of the solution, for both the symmetric and non--symmetric versions of DGFEM

Greedy approximation of high-dimensional Ornstein-Uhlenbeck operators with unbounded drift

We investigate the convergence of a nonlinear approximation method introduced by Ammar et al. (cf. J. Non-Newtonian Fluid Mech. 139:153--176, 2006) for the numerical solution of high-dimensional Fokker--Planck equations featuring in Navier--Stokes--Fokker--Planck systems that arise in kinetic models of dilute polymers. In the case of Poisson's equation on a rectangular domain in $\mathbb{R}^2$, subject to a homogeneous Dirichlet boundary condition, the mathematical analysis of the algorithm was carried out recently by Le Bris, Leli\`evre and Maday (Const. Approx. 30: 621--651, 2009), by exploiting its connection to greedy algorithms from nonlinear approximation theory explored, for example, by DeVore and Temlyakov (Adv. Comput. Math. 5:173--187, 1996); hence, the variational version of the algorithm, based on the minimization of a sequence of Dirichlet energies, was shown to converge. In this paper, we extend the convergence analysis of the pure greedy and orthogonal greedy algorithms considered by Le Bris, Leli\`evre and Maday to the technically more complicated case where the Laplace operator is replaced by a high-dimensional Ornstein--Uhlenbeck operator with unbounded drift, of the kind that appears in Fokker--Planck equations that arise in bead-spring chain type kinetic polymer models with finitely extensible nonlinear elastic potentials, posed on a high-dimensional Cartesian product configuration space D = D_1 x ... x D_N contained in $\mathbb{R}^{N d}$, where each set D_i, i=1,...,N, is a bounded open ball in $\mathbb{R}^d$, d = 2, 3

Discontinuous Galerkin finite element approximation of non-divergence form elliptic equations with Cordes coefficients

Non-divergence form elliptic equations with discontinuous coefficients do not generally posses a weak formulation, thus presenting an obstacle to their numerical solution by classical finite element methods. We propose a new $hp$-version discontinuous Galerkin finite element method for a class of these problems that satisfy the Cordes condition. It is shown that the method exhibits a convergence rate that is optimal with respect to the mesh size $h$ and suboptimal with respect to the polynomial degree $p$ by only half an order. Numerical experiments demonstrate the accuracy of the method and illustrate the potential of exponential convergence under $hp$-refinement for problems with discontinuous coefficients and nonsmooth solutions