An approximate Jacobian nonlinear solver for multiphase flow and transport

A block-diagonal preconditioner with the minimal residual method and an approximate block-factorization preconditioner with the generalized minimal residual method are developed for Hu-Zhang mixed finite element methods for linear elasticity. They are based on a new stability result for the saddle point system in mesh-dependent norms. The mesh-dependent norm for the stress corresponds to the mass matrix which is easy to invert while for the displacement it is spectral equivalent to the Schur complement. A fast auxiliary space preconditioner based on the H 1 H^1 -conforming linear element of the linear elasticity problem is then designed for solving the Schur complement. For both diagonal and triangular preconditioners, it is proved that the conditioning numbers of the preconditioned systems are bounded above by a constant independent of both the crucial Lamé constant and the mesh size. Numerical examples are presented to support theoretical results. As byproducts, a new stabilized low order mixed finite element method is proposed and analyzed and superconvergence results for the Hu-Zhang element are obtained.

In this article, an approach for the efficient numerical solution of multi-species reactive transport problems in porous media is described. The objective of this approach is to reformulate the given system of partial and ordinary differential equations (PDEs, ODEs) and algebraic equations (AEs), describing local equilibrium, in such a way that the couplings and nonlinearities are concentrated in a rather small number of equations, leading to the decoupling of some linear partial differential equations from the nonlinear system. Thus, the system is handled in the spirit of a global implicit approach (one step method) avoiding operator splitting techniques, solved by Newton’s method as the basic algorithmic ingredient. The reduction of the problem size helps to limit the large computational costs of numerical simulations of such problems. If the model contains equilibrium precipitation-dissolution reactions of minerals, then these are considered as complementarity conditions and rewritten as semismooth equations, and the whole nonlinear system is solved by the semismooth Newton method.

Numerical modelling of mass transport problems in porous media: A review

Low order nonconforming mixed finite element method for nonstationary incompressible Navier-Stokes equations

This paper proposes a finite volume–Yeh’s finite-element–multiscale finite-element model (FVYMSFEM) for solute transport simulations, which is extended from its elliptic groundwater-flow eq...

We present version 3 of the open-source simulator for flow and transport processes in porous media DuMux. DuMux is based on the modular C++ framework Dune (Distributed and Unified Numerics Environment) and is developed as a research code with a focus on modularity and reusability. We describe recent efforts in improving the transparency and efficiency of the development process and community-building, as well as efforts towards quality assurance and reproducible research. In addition to a major redesign of many simulation components in order to facilitate setting up complex simulations in DuMux, version 3 introduces a more consistent abstraction of finite volume schemes. Finally, the new framework for multi-domain simulations is described, and three numerical examples demonstrate its flexibility.

We consider adaptive discontinuous Galerkin (DG) methods for solving reactive transport problems in porous media. To guide anisotropic and dynamic mesh adaptation, a posteriori error estimators based on solving local problems are established. These error estimators are efficient to compute and effective to capture local phenomena, and they apply to all the four primal DG schemes, namely, symmetric interior penalty Galerkin, nonsymmetric interior penalty Galerkin, incomplete interior penalty Galerkin, and the Oden–Babuska-Baumann version of DG. Numerical results are provided to illustrate the effectiveness of the proposed error estimators.

Simulation of reactive transport in porous media using radial point collocation method

In this article, we investigate a priori error estimates for the optimal control problems governed by elliptic equations using higher order variational discretization and mixed finite element methods. The state and the co-state are approximated by the order k Raviart-Thomas mixed finite element spaces and the control is not discreted. A priori error estimates for the higher order variational discretization and mixed finite element approximation of control problems are obtained. Finally, we present some numerical examples which confirm our theoretical results.Mathematics Subject Classification 1991: 49J20; 65N30.

The melting process with moving boundary for mixture consisting of two fluids

p-version of mixed finite element methods for Stokes-like problems

A combined mixed and discontinuous Galerkin method for compressible miscible displacement problem in porous media

Parallel multi-stage preconditioners with adaptive setup for the black oil model

A combined method consisting of the mixed finite element method for flow and the local discontinuous Galerkin method for transport is introduced for the one-dimensional coupled system of incompressible miscible displacement problem. Optimal error estimates in L ∞(0, T;L 2) for concentration c, in L 2(0, T; L 2) for c x and L ∞(0, T;L 2) for velocity u are derived. The main technical difficulties in the analysis include the treatment of the inter-element jump terms which arise from the discontinuous nature of the numerical method, the nonlinearity, and the coupling of the models. Numerical experiments are performed to verify the theoretical results. Finally, we apply this method to the one-dimensional compressible miscible displacement problem and give the numerical experiments to confirm the efficiency of the scheme.

The paper deals with the analysis of three-dimensional incompressible flows of generalized Newtonian liquids in corotating twin-screw equipment where the fluid-filled region is rather complex and changes in time as the screws rotate. Taking notice of the fact that the flow fields are periodic in time and, simultaneously, periodic in axial direction, a mixed finite element method is applied for the sake of discretization. The resulting systems of nonlinear algebraic equations are treated by a Newton iteration where the linear subproblems with indefinite sparse coefficient matrices are solved by applying a modified Uzawa algorithm. By simulating different working conditions, the screw characteristics are extracted numerically, and the influence of Deborah number is demonstrated. Selected flow fields are visualized grafically illustrating the intermeshing screw operation principle and demonstrating certain symmetries connected with the kinematic reversibility of nonlinear viscous creeping flows.