Abstract. In this paper, we present a new optimal interpolation error estimate in L p norm (1 ≤ p ≤ ∞) for finite element simplicial meshes in any spatial dimension. A sufficient condition for a mesh to be nearly optimal is that it is quasi-uniform under a new metric defined by a modified Hessian matrix of the function to be interpolated. We also give new functionals for the global moving mesh method and obtain optimal monitor functions from the viewpoint of minimizing interpolation error in the L p norm. Some numerical examples are also given to support the theoretical estimates.

In this paper we study the a priori error estimates of finite element method for the system of time-dependent Poisson-Nernst-Planck equations, and for the first time, we obtain its optimal error estimates in L ∞ (H 1) and L 2 (H 1) norms, and suboptimal error estimates in L ∞ (L 2) norm, with linear element, and optimal error estimates in L ∞ (L 2) norm with quadratic or higher-order element, for both semi-and fully discrete finite element approximations. Numerical experiments are also given to validate the theoretical results.

Extreme mass ratio binary systems, binaries involving stellar mass objects orbiting massive black holes, are considered to be a primary source of gravitational radiation to be detected by the spacebased interferometer LISA. The numerical modelling of these binary systems is extremely challenging because the scales involved expand over several orders of magnitude. One needs to handle large wavelength scales comparable to the size of the massive black hole and, at the same time, to resolve the scales in the vicinity of the small companion where radiation reaction effects play a crucial role. Adaptive finite element methods, in which quantitative control of errors is achieved automatically by finite element mesh adaptivity based on posteriori error estimation, are a natural choice that has great potential for achieving the high level of adaptivity required in these simulations. To demonstrate this, we present the results of simulations of a toy model, consisting of a point-like source orbiting a black hole under the action of a scalar gravitational field.

[1] Particle-based models and continuum models have been developed to quantify mixinglimited bimolecular reactions for decades. Effective model parameters control reaction kinetics, but the relationship between the particle-based model parameter (such as the interaction radius R) and the continuum model parameter (i.e., the effective rate coefficient K f ) remains obscure. This study attempts to evaluate and link R and K f for the second-order bimolecular reaction in both the bulk and the sharp-concentration-gradient (SCG) systems. First, in the bulk system, the agent-based method reveals that R remains constant for irreversible reactions and decreases nonlinearly in time for a reversible reaction, while mathematical analysis shows that K f transitions from an exponential to a power-law function. Qualitative link between R and K f can then be built for the irreversible reaction with equal initial reactant concentrations. Second, in the SCG system with a reaction interface, numerical experiments show that when R and K f decline as t À1/2 (for example, to account for the reactant front expansion), the two models capture the transient power-law growth of product mass, and their effective parameters have the same functional form. Finally, revisiting of laboratory experiments further shows that the best fit factor in R and K f is on the same order, and both models can efficiently describe chemical kinetics observed in the SCG system. Effective model parameters used to describe reaction kinetics therefore may be linked directly, where the exact linkage may depend on the chemical and physical properties of the system. Citation: Zhang, Y., C. Papelis, P. Sun, and Z. Yu (2013), Evaluation and linking of effective parameters in particle-based models and continuum models for mixing-limited bimolecular reactions, Water Resour. Res., 49,

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