Abstract

The present article describes a simple element-driven strategy for the conforming refinement of simplicial finite element meshes in a distributed environment. The proposed algorithm is effective both for local adaptive refinement and for the division of all the elements within an existing mesh. We aim to provide sufficient detail to allow the practical implementation of the algorithm, which can be coded with minimal effort provided that a distributed linear algebra library is available. The proposed refinement strategy is composed of three basic components: a global splitting strategy, an elemental splitting procedure and an error estimation technique, which are combined so to guarantee obtaining a conformant refined mesh. A number of benchmark examples show the capabilities of the proposed method. Error is estimated for the incompressible fluid-flow benchmarks using a novel indicator based on the computation of the sub-scale velocity.

Abstract

This article investigates an explicit a-posteriori error estimator for the finite element approximation of the compressible Navier–Stokes equations. The proposed methodology employs the Variational Multi-Scale framework, and specifically, the idea is to use the variational subscales to estimate the error. These subscales are defined to be orthogonal to the finite element space, dynamic and non-linear, and both the subscales in the interior of the element and on the element boundaries are considered. Another particularity of the model is that we define some norms that lead to a dimensionally consistent measure of the compressible flow solution error inside each element; a scaled L2-norm, and the calculation of a physical entropy measure, are both studied in this work. The estimation of the error is used to drive the adaptive mesh refinement of several compressible flow simulations. Numerical results demonstrate good accuracy of the local error estimate and the ability to drive the adaptative mesh refinement to minimize the error through the computational domain.

Abstract

We propose a novel approach for flow simulations in fractured porous media based on an equi-dimensional representation of the fractures and a standard continuous finite element (FE) method. We employ an adaptive mesh refinement strategy to automatically adjust an initial regular mesh to any fracture distribution with a desired accuracy. The proposed approach is easily implementable in any FE software, does not involve the coupling of different discretizations, and provides symmetric and positive definite stiffness matrices. We provide a systematic validation of our method and we show that it can be used to simulate fluid flow for fracture networks of realistic complexity.

Abstract

We present highly scalable parallel distributed-memory algorithms and associated data structures for a generic finite element framework that supports h-adaptivity on computational domains represented as multiple connected adaptive trees—forest-of-trees—, thus providing multi-scale resolution on problems governed by partial differential equations.The framework is grounded on a rich representation of the adaptive mesh suitable for generic finite elements that is built on top of a low-level, light-weight forest-oftrees data structure handled by a specialized, highly parallel adaptive meshing engine. Along the way, we have identified the requirements that the forest-of-trees layer must fulfill to be coupled into our framework. Essentially, it must be able to describe neighboring relationships between cells in the adapted mesh (apart from hierarchical relationships) across the lower-dimensional objects at the boundary of the cells. Atop this two-layered mesh representation, we build the rest of data structures required for the numerical integration and assembly of the discrete system of linear equations.We consider algorithms that are suitable for both subassembled and fully-assembled distributed data layouts of linear system matrices. The proposed framework has been implemented within the FEMPAR scientific software library, using p4est as a practical forest-of-octrees demonstrator. A comprehensive strong scaling study of this implementation when applied to Poisson and Maxwell problems reveals remarkable scalability up to 32.2K CPU cores and 482.2M degrees of freedom. Besides, the implementation in FEMPAR of the proposed approach is up to 2.6 and 3.4 times faster than the state-of-the-art deal.II finite element software in the h-adaptive approximation of a Poisson problem with firstand second-order Lagrangian finite elements, respectively (excluding the linear solver step from the comparison).

Abstract

Abstract