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

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 $1\leq s\leq\min\set{p+1,k}$

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

Solving Linearized Equations of the NN-body Problem Using the Lie-integration Method

Several integration schemes exits to solve the equations of motion of the $N$-body problem. The Lie-integration method is based on the idea to solve ordinary differential equations with Lie-series. In the 1980s this method was applied for the $N$-body problem by giving the recurrence formula for the calculation of the Lie-terms. The aim of this works is to present the recurrence formulae for the linearized equations of motion of $N$-body systems. We prove a lemma which greatly simplifies the derivation of the recurrence formulae for the linearized equations if the recurrence formulae for the equations of motions are known. The Lie-integrator is compared with other well-known methods. The optimal step size and order of the Lie-integrator are calculated. It is shown that a fine-tuned Lie-integrator can be 30%-40% faster than other integration methods.Comment: accepted for publication in MNRAS (13 pages, 4 figures); see http://cm.elte.hu/lie (cm.elte.hu/lie) for softwar

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

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

A-priori analysis of the quasicontinuum method in one dimension

The quasicontinuum method is a coarse-graining technique for reducing the complexity of atomistic simulations in a static and quasistatic setting. In this paper we give an a-priori error analysis for the quasicontinuum method in one dimension. We consider atomistic models with Lennard-Jones type long range interactions and a practical QC formulation.\ud \ud First, we prove the existence, the local uniqueness and the stability with respect to discrete W1,∞-norm of elastic and fractured atomistic solutions. We then used a fixed point technique to prove the existence of quasicontinuum approximation which satisfies an optimal a-priori error bound.\ud \ud The first-named author acknowledges the financial support received from the European research project HPRN-CT-2002-00284: New Materials, Adaptive Systems and their Nonlinearities. Modelling, Control and Numerical Simulation, and the kind hospitality of Carlo Lovadina (University of Pavia)