Adaptive Time Stepping, Linearization and a Posteriori Error Control for Multiphase Flow with Wells

We present in this paper a-posteriori error estimators for multiphase flow with singular well sources. The estimators are fully and locally computable, distinguish the various error components, and target the singular effects of wells. On the basis of these estimators we design an adaptive fully-implicit solver that yields optimal nonlinear iterations and efficient time-stepping, while maintaining the accuracy of the solution. A key point is that the singular nature of the solution in the near-well region is explicitly captured and efficiently estimated using the adequate norms. Numerical experiments illustrate the efficiency of our estimates and the performance of the adaptive algorithm.

Superconvergent gradient recovery for virtual element methods

Virtual element method is a new promising finite element method using general polygonal meshes. Its optimal a priori error estimates are well established in the literature. In this paper, we take a different viewpoint. We try to uncover the superconvergent property of virtual element methods by doing some local post-processing only on the degrees of freedom. Using the linear virtual element method as an example, we propose a universal gradient recovery procedure to improve the accuracy of gradient approximation for numerical methods using general polygonal meshes. Its capability of serving as a posteriori error estimators in adaptive computation is also investigated. Compared to the existing residual-type a posteriori error estimators for the virtual element methods, the recovery-type a posteriori error estimator based on the proposed gradient recovery technique is much simpler in implementation and it is asymptotically exact. A series of benchmark tests are presented to numerically illustrate the superconvergence of recovered gradient and validate the asymptotic exactness of the recovery-based a posteriori error estimator.

Adaptive finite element methods for sparse PDE-constrained optimization

We propose and analyse reliable and efficient a posteriori error estimators for an optimal control problem that involves a nondifferentiable cost functional, the Poisson problem as state equation and control constraints. To approximate the solution to the state and adjoint equations we consider a piecewise linear finite element method, whereas three different strategies are used to approximate the control variable: piecewise constant discretization, piecewise linear discretization and the so-called variational discretization approach. For the first two aforementioned solution techniques we devise an error estimator that can be decomposed as the sum of four contributions: two contributions that account for the discretization of the control variable and the associated subgradient and two contributions related to the discretization of the state and adjoint equations. The error estimator for the variational discretization approach is decomposed only in two contributions that are related to the discretization of the state and adjoint equations. On the basis of the devised a posteriori error estimators, we design simple adaptive strategies that yield optimal rates of convergence for the numerical examples that we perform.