Abstract. We analyze the simplest and most standard adaptive finite element method (AFEM), with any polynomial degree, for general second order linear, symmetric elliptic operators. As is customary in practice, the AFEM marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property. We prove that the AFEM is a contraction, for the sum of the energy error and the scaled error estimator, between two consecutive adaptive loops. This geometric decay is instrumental to derive the optimal cardinality of the AFEM. We show that the AFEM yields a decay rate of the energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity.

Ultraintense laser pulses with a few-cycle rising edge are ideally suited to accelerating ions from ultrathin foils, and achieving such pulses in practice represents a formidable challenge. We show that such pulses can be obtained using sufficiently strong and well-controlled relativistic nonlinearities in spatially well-defined near-critical-density plasmas. The resulting ultraintense pulses with an extremely steep rising edge give rise to significantly enhanced carbon ion energies consistent with a transition to radiation pressure acceleration.

We study an adaptive finite element method for the p-Laplacian like PDEs using piecewise linear, continuous functions. The error is measured by means of the quasi norm of Barrett and Liu. We provide residual based error estimators without a gap between the upper and lower bound. We show linear convergence of the algorithm which is similar to the one of Morin, Nochetto, and Siebert. All results are obtained without extra marking for the oscillation.

Table-top laser–plasma ion accelerators have many exciting applications, many of which require ion beams with simultaneous narrow energy spread and high conversion efficiency. However, achieving these requirements has been elusive. Here we report the experimental demonstration of laser-driven ion beams with narrow energy spread and energies up to 18 MeV per nucleon and ∼5% conversion efficiency (that is 4 J out of 80-J laser). Using computer simulations we identify a self-organizing scheme that reduces the ion energy spread after the laser exits the plasma through persisting self-generated plasma electric (∼1012 V m−1) and magnetic (∼104 T) fields. These results contribute to the development of next generation compact accelerators suitable for many applications such as isochoric heating for ion-fast ignition and producing warm dense matter for basic science.

We develop the analysis of finite element approximations of implicit power-law-like models for viscous incompressible fluids. The Cauchy stress and the symmetric part of the velocity gradient in the class of models under consideration are related by a, possibly multivalued, maximal monotone r-graph with 1 < r < ∞. Using a variety of weak compactness techniques, including Chacon's biting lemma and Young measures, we show that a subsequence of the sequence of finite element solutions converges to a weak solution of the problem as the finite element discretization parameter h tends to 0. A key new technical tool in our analysis is a finite element counterpart of the Acerbi-Fusco Lipschitz truncation of Sobolev functions.