Volume 32, Issue 2

In the present paper, we examine a Crouzeix–Raviart approximation for non-linear partial differential equations having a ( p , δ )-structure for some p ∈ (1, ∞) and δ ⩾ 0. We establish a priori error estimates, which are optimal for all p ∈ (1, ∞) and δ ⩾ 0, medius error estimates, i.e., best-approximation results, and a primal–dual a posteriori error estimate, which is both reliable and efficient. The theoretical findings are supported by numerical experiments.

We propose high order conforming and nonconforming immersed hybridized difference (IHD) methods in two and three dimensions for elliptic interface problems. Introducing the virtual to real transformation (VRT), we could obtain a systematic and unique way of deriving arbitrary high order methods in principle. The optimal number of collocating points for imposing interface conditions is proved, and a unique way of constructing the VRT is suggested. Numerical experiments are performed in two and three dimensions. Numerical results achieving up to the 6th order convergence in the L 2 -norm are presented for the two dimensional case, and a three dimensional example with a 4th order convergence is presented.

This paper presents a posteriori error estimate for the weak Galerkin (WG) finite element method used to solve H ( curl )-elliptic problems. Firstly, we introduce a WG method for solving H ( curl )-elliptic problems and a corresponding residual type error estimator without a stabilization term. Secondly, we establish the reliability of the error estimator by demonstrating that the stabilization term is controlled by the error estimator. We also evaluate the efficiency of the error estimator using standard bubble functions. Finally, we present some numerical results to show the performances of the error estimator in both uniform and adaptive meshes.

The Keller–Segel–Navier–Stokes system governs chemotaxis in liquid environments. This system is to be solved for the organism and chemoattractant densities and for the fluid velocity and pressure. It is known that if the total initial organism density mass is below 2 π there exist globally defined generalised solutions, but what is less understood is whether there are blow-up solutions beyond such a threshold and its optimality. Motivated by this issue, a numerical blow-up scenario is investigated. Approximate solutions computed via a stabilised finite element method founded on a shock capturing technique are such that they satisfy a priori bounds as well as lower and L 1 ( Ω ) bounds for the organism and chemoattractant densities. In particular, these latter properties are essential in detecting numerical blow-up configurations, since the non-satisfaction of these two requirements might trigger numerical oscillations leading to non-realistic finite-time collapses into persistent Dirac-type measures. Our findings show that the existence threshold value 2 π encountered for the organism density mass may not be optimal and hence it is conjectured that the critical threshold value 4 π may be inherited from the fluid-free Keller–Segel equations. Additionally it is observed that the formation of singular points can be neglected if the fluid flow is intensified.