We develop an adaptive finite element method for island dynamics in epitaxial growth. We study a step-flow model, which consists of an adatom (adsorbed atom) diffusion equation on terraces of different height; thermodynamic boundary conditions on terrace boundaries including anisotropic line tension; and the normal velocity law for the motion of such boundaries determined by a two-sided flux, t...

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We provide a common framework for the understanding, comparison, and analysis of several discontinuous Galerkin methods that have been proposed for the numerical treatment of elliptic problems. This class includes the recently introduced methods of Bassi and Rebay (together with the variants proposed by Brezzi, Manzini, Marini, Pietra and Russo), the local discontinuous Galerkin methods of Cock...

This paper considers a finite element approximation of the Dirichlet problem for a second order self-adjoint elliptic equation, A u =f, in a region f 2 c N " (n=2 or 3) by the boundary penalty method. If the finite element space defined over D h, a union of elements, has approximation power h ~ in the L 2 norm, then (i) for f2=D h convex polyhedral, we show that choosing the penalty parameter e...

The symmetric interior penalty (SIP) method on graded meshes and its fast solution by multigrid methods are studied in this paper. We obtain quasi-optimal error estimates in both the energy norm and the L2 norm for the SIP method, and prove uniform convergence of the W -cycle multigrid algorithm for the resulting discrete problem. The performance of these methods is illustrated by numerical res...

Computing meaningful alternative routes in a road network is a complex problem – already giving a clear definition of a best alternative seems to be impossible. Still, multiple methods [1, 2, 4, 17, 18] describe how to compute reasonable alternative routes, each according to their own quality criteria. Among these methods, the penalty method has received much less attention than the via-node or...

This note considers possible arrangements of the sectors of a generalised dartboard. The sum of the pth powers of the absolute differences of the numbers on adjacent sectors is introduced as a penalty cost function and a string reversal algorithm is used to determine all arrangements that maximise the penalty, for any p ≥ 1. The maximum value of the penalty function for p = 1 is well known in t...

In this paper, we give a sufficient condition for the asymptotic convergence of penalty trajectories in convex programming with multiple solutions. We show that, for a wide class of penalty methods, the associated optimal trajectory converges to a particular solution of the original problem, characterized through a minimization selection principle. Our main assumption for this convergence resul...

We introduce stable multi-domain spectral penalty methods suitable for solving fractional partial differential equations with fractional derivatives of any order. The fractional derivative is approximated in each sub domain using orthogonal polynomials and stability of the scheme is achieved through weakly imposed boundary and interface conditions by using a penalty term. We discuss accuracy an...

In this paper, we consider the surplus process of the classical continuous time risk model containing an independent diffusion (Wiener) process. We generalize the defective renewal equation for the expected discounted function of a penalty at the time of ruin in Garber and Landry [Insurance: Math. Econ. 22 (1998) 263]. Then an asymptotic formula for the expected discounted penalty function is p...