Dedicated to Carsten Carstensen on the occasion of his 50th birthday.Abstract. Newest vertex bisection (NVB) is a popular local mesh-refinement strategy for regular triangulations which consist of simplices. For the 2D case, we prove that the meshclosure step of NVB, which preserves regularity of the triangulation, is quasi-optimal and that the corresponding L 2 -projection onto lowest-order Courant finite elements (P1-FEM) is always H 1 -stable. Throughout, no additional assumptions on the initial triangulation are imposed. Our analysis thus improves results of Binev, Dahmen & DeVore (Numer. Math. 97, 2004), Carstensen (Constr. Approx. 20, 2004), and Stevenson (Math. Comp. 77, 2008 in the sense that all assumptions of their theorems are removed. Consequently, our results relax the requirements under which adaptive finite element schemes can be mathematically guaranteed to convergence with quasi-optimal rates.

Abstract. We construct and analyze strong stability preserving implicit-explicit Runge-Kutta methods for the time integration of models of flow and radiative transport in astrophysical applications. It turns out that in addition to the optimization of the region of absolute monotonicity, other properties of the methods are crucial for the success of the simulations as well. The models in our focus dictate to also take into account the step-size limits associated with dissipativity and positivity of the stiff parabolic terms which represent transport by diffusion. Another important property is uniform convergence of the numerical approximation with respect to different stiffness of those same terms. Furthermore, it has been argued that in the presence of hyperbolic terms, the stability region should contain a large part of the imaginary axis even though the essentially non-oscillatory methods used for the spatial discretization have eigenvalues with a negative real part. Hence, we construct several new methods which differ from each other by taking various or even all of these constraints simultaneously into account. It is demonstrated for the problem of double-diffusive convection that the newly constructed schemes provide a significant computational advantage over methods from the literature. They may also be useful for general problems which involve the solution of advectiondiffusion equations, or other transport equations with similar stability requirements.

For the simple layer potential V that is associated with the 3D Laplacian, we consider the weakly singular integral equation V φ = f. This equation is discretized by the lowest order Galerkin boundary element method. We prove convergence of an h-adaptive algorithm that is driven by a weighted residual error estimator. Moreover, we identify the approximation class for which the adaptive algorithm converges quasi-optimally with respect to the number of elements. In particular, we prove that adaptive mesh refinement is superior to uniform mesh refinement.

-We prove convergence and quasi-optimality of a lowest-order adaptive boundary element method for a weakly-singular integral equation in 2D. The adaptive mesh-refinement is driven by the weighted-residual error estimator. By proving that this estimator is not only reliable, but under some regularity assumptions on the given data also efficient on locally refined meshes, we characterize the approximation class in terms of the Galerkin error only. In particular, this yields that no adaptive strategy can do better, and the weighted-residual error estimator is thus an optimal choice to steer the adaptive mesh-refinement. As a side result, we prove a weak form of the saturation assumption. 2010 Mathematical subject classification: 65N30, 65N15, 65N38.

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