Hierarchical Riesz bases for Hs(Omega), 1 < s < 5/2

On arbitrary polygonal domains $Omega subset RR^2$, we construct $C^1$ hierarchical Riesz bases for Sobolev spaces $H^s(Omega)$. In contrast to an earlier construction by Dahmen, Oswald, and Shi (1994), our bases will be of Lagrange instead of Hermite type, by which we extend the range of stability from $s in (2,frac{5}{2})$ to $s in (1,frac{5}{2})$. Since the latter range includes $s=2$, with respect to the present basis, the stiffness matrices of fourth-order elliptic problems are uniformly well-conditioned

An optimal adaptive wavelet method for First Order System Least Squares

In this paper, it is shown that any well-posed 2nd order PDE can be reformulated as a well-posed first order least squares system. This system will be solved by an adaptive wavelet solver in optimal computational complexity. The applications that are considered are second order elliptic PDEs with general inhomogeneous boundary conditions, and the stationary Navier-Stokes equations.Comment: 40 page

A convenient inclusion of inhomogeneous boundary conditions in minimal residual methods

Inhomogeneous essential boundary conditions can be appended to a well-posed PDE to lead to a combined variational formulation. The domain of the corresponding operator is a Sobolev space on the domain $\Omega$ on which the PDE is posed, whereas the codomain is a Cartesian product of spaces, among them fractional Sobolev spaces of functions on $\partial\Omega$. In this paper, easily implementable minimal residual discretizations are constructed which yield quasi-optimal approximation from the employed trial space, in which the evaluation of fractional Sobolev norms is fully avoided.Comment: Accepted for publication in CMA

An optimal adaptive Fictitious Domain Method

We consider a Fictitious Domain formulation of an elliptic partial differential equation and approximate the resulting saddle-point system using an inexact preconditioned Uzawa iterative algorithm. Each iteration entails the approximation of an elliptic problems performed using adaptive finite element methods. We prove that the overall method converges with the best possible rate and illustrate numerically our theoretical findings

Somalia and the Pirates. ESF Working Paper No. 33, 18 December 2009

Piracy is defined by The Hague Centre for Strategic Studies as an "act of boarding or attempting to board any ship with the apparent intent to commit theft or any other crime and with the apparent intent or capability to use force in furtherance of that act." And it is estimated that from 1995 to 2009, around 730 persons were killed or are presumed dead, approximately 3,850 seafarers were held hostage, around 230 were kidnapped and ransomed, nearly 800 were seriously injured and hundreds more were threatened with guns and knives. (See paper by Rob de Wijk). In November 2009, CEPS held a European Security Forum seminar, in collaboration with the Institute for Strategic Studies, the Centre for the Democratic Control of Armed Forces and the Geneva Centre for Security Policy, to focus on the issue of Somalia and the Pirates, chaired by Francois Heisbourg. Four eminent specialists in this field: David Anderson, Rob de Wijk, Steven Haines and Jonathon Stevenson looked at the links with Somalia, and the historical, legal, political and security dimensions of the troubling success of piracy in today’s world. Their conclusions and recommendations for future action are brought together in this ESF 33 Working Paper

Adaptive Spectral Galerkin Methods with Dynamic Marking

The convergence and optimality theory of adaptive Galerkin methods is almost exclusively based on the D\"orfler marking. This entails a fixed parameter and leads to a contraction constant bounded below away from zero. For spectral Galerkin methods this is a severe limitation which affects performance. We present a dynamic marking strategy that allows for a super-linear relation between consecutive discretization errors, and show exponential convergence with linear computational complexity whenever the solution belongs to a Gevrey approximation class.Comment: 20 page

A saturation property for the spectral-Galerkin approximation of a Dirichlet problem in a square

Both practice and analysis of adaptive $p$-FEMs and $hp$-FEMs raise the question what increment in the current polynomial degree $p$ guarantees a $p$-independent reduction of the Galerkin error. We answer this question for the $p$-FEM in the simplified context of homogeneous Dirichlet problems for the Poisson equation in the two dimensional unit square with polynomial data of degree $p$. We show that an increment proportional to $p$ yields a $p$-robust error reduction and provide computational evidence that a constant increment does not

On p-Robust Saturation for hp-AFEM

We consider the standard adaptive finite element loop SOLVE, ESTIMATE, MARK, REFINE, with ESTIMATE being implemented using the $p$-robust equilibrated flux estimator, and MARK being D\"orfler marking. As a refinement strategy we employ $p$-refinement. We investigate the question by which amount the local polynomial degree on any marked patch has to be increase in order to achieve a $p$-independent error reduction. The resulting adaptive method can be turned into an instance optimal $hp$-adaptive method by the addition of a coarsening routine