The DUNE-ALUGrid Module

In this paper we present the new DUNE-ALUGrid module. This module contains a major overhaul of the sources from the ALUgrid library and the binding to the DUNE software framework. The main changes include user defined load balancing, parallel grid construction, and an redesign of the 2d grid which can now also be used for parallel computations. In addition many improvements have been introduced into the code to increase the parallel efficiency and to decrease the memory footprint. The original ALUGrid library is widely used within the DUNE community due to its good parallel performance for problems requiring local adaptivity and dynamic load balancing. Therefore, this new model will benefit a number of DUNE users. In addition we have added features to increase the range of problems for which the grid manager can be used, for example, introducing a 3d tetrahedral grid using a parallel newest vertex bisection algorithm for conforming grid refinement. In this paper we will discuss the new features, extensions to the DUNE interface, and explain for various examples how the code is used in parallel environments.Comment: 25 pages, 11 figure

Mesh refinement for parallel-adaptive FEM : theory and implementation

We investigate parallel adaptive grid refinement and focus in particular on hierarchically adaptive, parallel and conforming simplicial grids that use Newest Vertex Bisection (NVB) as their refinement strategy. One challenge of NVB is its applicability to arbitrary simplex grids, which is not possible with the current compatibility condition. We define a novel, more natural weak compatibility condition for the initial grid and show that using this condition the iterative refinement algorithm terminates using NVB. We design an algorithm to relabel an arbitrary d-dimensional simplicial grid to fulfil this weak compatibility condition. The algorithm is of complexity O(n), where n is the number of elements in the grid. We also consider NVB on partitioned grids for parallel computing. Another challenge is that refinement may propagate over partition boundaries. This is resolved by adding an outer loop to the refinement algorithm, that requires global communication. We prove that the amount of global communication needed and the number of outer iterations in the refinement propagation to reach a conforming situation is bounded. We extend the grid manager DUNE-ALUGrid to provide parallel, adaptive, conforming 2d grids. Furthermore we develop the software package DUNE-ACFem which is able to conveniently describe mathematical problems within efficient C++ code. We demonstrate the utility of DUNE-ACFEM and DUNE-ALUGrid at the problem of noise removal on images with adaptive finite elements

A primal-dual finite element method for scalar and vectorial total variation minimization

Based on the Fenchel duality we build a primal-dual framework for minimizing a general functional consisting of a combined L1 and L2 data-fidelity term and a scalar or vectorial total variation regularisation term. The minimization is performed over the space of functions of bounded variations and appropriate discrete subspaces. We analyze the existence and uniqueness of solutions of the respective minimization problems. For computing a numerical solution we derive a semi-smooth Newton method on finite element spaces and highlight applications in denoising, inpainting and optical flow estimation

Accuracy of fully coupled and sequential approaches for modeling hydro- and geomechanical processes

<jats:title>Abstract</jats:title><jats:p>Subsurface flow and geomechanics are often modeled with sequential approaches. This can be computationally beneficial compared with fully coupled schemes, while it requires usually compromises in numerical accuracy, at least when the sequential scheme is non-iterative. We discuss the influence of the choice of scheme on the numerical accuracy and the expected computational effort based on a comparison of a fully coupled scheme, a scheme employing a one-way coupling, and an iterative scheme using a fixed-stress split for two subsurface injection scenarios. All these schemes were implemented in the numerical simulator DuMu<jats:sup>x</jats:sup>. This study identifies conditions of problem settings where differences due to the choice of the model approach are as important as differences in geologic features. It is shown that in particular transient and multiphase flow, effects can be causing significant deviations between non-iterative and iterative sequential schemes, which might be in the same order of magnitude as geologic uncertainty. An iterated fixed-stress split has the same numerical accuracy as a fully coupled scheme but only for a certain number of iterations which might use up the computational advantage of solving two smaller systems of equations rather than a big monolithical one.</jats:p&gt