European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
10:40   MS25: Discretizations of mixed-dimensional PDEs (Part 2)
Chair: Marie Rognes
10:40
25 mins
An Adaptive Penalty Method for Inequality Constrained Minimization Problems
Wietse Boon, Jan Nordbotten
Abstract: The primal-dual active set method is observed to be the limit of a sequence of penalty formulations. Using this perspective, we propose a penalty method that adaptively becomes the active set method as the residual of the iterate decreases. The adaptive penalty method (APM) therewith combines the main advantages of both methods, namely the ease of implementation of penalty methods and the exact imposition of inequality constraints inherent to the active set method. The scheme can be considered a quasi-Newton method in which the Jacobian is approximated using a penalty parameter. This spatially varying parameter is chosen at each iteration by solving an auxiliary problem.
11:05
25 mins
Coupled Flow and Mechanics in a 3D Porous Media with Line Sources
Nadia S. Taki, Ingeborg Gjerde
Abstract: In this paper, we consider the numerical approximation of the quasi-static, linear Biot model in a 3D domain Ω when the right-hand side of the flow equation is concentrated on a 1D line source δΛ. This model is of interest in the context of medicine, where it can be used to model flow and deformation through vascularized tissue. The model itself is challenging to approximate as the line source induces the pressure and flux solution to be singular. To overcome this, we here combine two methods: (i) a fixed-stress splitting scheme to decouple the flow and mechanics equations and (ii) a singularity removal method for the pressure and flux variables. The singularity removal is based on a splitting of the solution into a lower regularity term capturing the solution singularities and a higher regularity term denoted the remainder. With this in hand, the flow equations can now be reformulated so that they are posed with respect to the remainder terms. The reformulated system is then approximated using the fixed-stress splitting scheme. We conclude by showing the results for a test case simulating flow through vascularized tissue. Here, the numerical method is found to converge optimally using lowest-order elements for the spatial discretization.
11:30
25 mins
A posteriori model error analysis of 3D-1D coupled PDEs
Federica Laurino, Stefano Brambilla, Paolo Zunino
Abstract: The objective of this work is to extend the model reduction technique for coupled 3D-1D elliptic PDEs, previously proposed by the authors, with an a posteriori analysis of the model error, defined as the difference between the solutions of the reference and reduced problem. More precisely, we introduce an estimator for an user-defined functional of the error, computed using a duality approach. This results particularly useful since it allows to localize the model error on the computational mesh and to investigate the reliability of the model reduction approach.
11:55
25 mins
A stabilized cut discontinuous Galerkin framework for mixed dimensional coupled problems
Andre Massing, Ceren Gürkan, Christoph Lehrenfeld, Fabian Heimann
Abstract: We develop both theoretically and practically a novel cut Discontinuous Galerkin framework (cutDG) by combining stabilization techniques from the cut finite element method [1] with the classical interior penalty discontinuous Galerkin methods (DG) for elliptic [2] and hyperbolic problems [3]. The resulting framework allows for the unified numerical treatment of a wide range of problems, including boundary and interface problems as well as surface [4,5] and mul- tidimensional, coupled surface-bulk and interface-bulk problems. The key idea is that the domains of interest such the surface or the bulk domain can be embedded into a background mesh in an unfitted manner. Using only a few abstract assumptions on the employed cutDG stabilization, we can establish geometrically robust optimal a priori error and condition number estimates irrespective of how the embedded geometry cuts the background mesh. Possible realizations of the cutDG stabilization are discussed. The theoretical properties are corroborated by a number of elliptic and hyperbolic prototype problems ranging from simple boundary values and surface problem to multidimensional coupled interface-bulk problems.