We analyze adaptive mesh-refining algorithms for conforming finite element discretizations of certain nonlinear second-order partial differential equations. We allow continuous polynomials of arbitrary but fixed polynomial order. The adaptivity is driven by the residual error estimator. We prove convergence even with optimal algebraic convergence rates. In particular, our analysis covers general linear second-order elliptic operators. Unlike prior works for linear nonsymmetric operators, our analysis avoids the interior node property for the refinement, and the differential operator has to satisfy a Gårding inequality only. If the differential operator is uniformly elliptic, no additional assumption on the initial mesh is posed.

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.

We develop and analyze an ultraweak variational formulation for a variant of the Kirchhoff-Love plate bending model. Based on this formulation, we introduce a discretization of the discontinuous Petrov-Galerkin type with optimal test functions (DPG). We prove wellposedness of the ultraweak formulation and quasi-optimal convergence of the DPG scheme.The variational formulation and its analysis require tools that control traces and jumps in H 2 (standard Sobolev space of scalar functions) and H(div div) (symmetric tensor functions with L 2 -components whose twice iterated divergence is in L 2 ), and their dualities. These tools are developed in two and three spatial dimensions. One specific result concerns localized traces in a dense subspace of H(div div). They are essential to construct basis functions for an approximation of H(div div).To illustrate the theory we construct basis functions of the lowest order and perform numerical experiments for a smooth and a singular model solution. They confirm the expected convergence behavior of the DPG method both for uniform and adaptively refined meshes.

-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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