Variational inequality approach to enforcing the non-negative constraint for advection–diffusion equations

Abstract

These figures depict the concentration profiles of the advection-diffusion equation under the Discontinuous Galerkin formulation (left) and the Discontinuous Galerkin formulation with bounded constraints enforced through a variational inequality (right). Only the regions where concentrations meet the threshold of 0.5 and above are shown. Abstract. Predictive simulations are crucial for the success of many subsurface applications, and it is highly desirable to obtain accurate non-negative solutions for transport equations in these numerical simulations. To this end, optimization-based methodologies based on quadratic programming (QP) have been shown to be a viable approach to ensuring discrete maximum principles and the non-negative constraint for anisotropic diffusion equations. In this paper, we propose a computational framework based on the variational inequality (VI) which can also be used to enforce important mathematical properties (e.g., maximum principles) and physical constraints (e.g., the non-negative constraint). We demonstrate that this framework is not only applicable to diffusion equations but also to non-symmetric advection-diffusion equations. An attractive feature of the proposed framework is that it works with with any weak formulation for the advection-diffusion equations, including single-field formulations, which are computationally attractive. A particular emphasis is placed on the parallel and algorithmic performance of the VI approach across large-scale and heterogeneous problems. It is also shown that QP and VI are equivalent under certain conditions. State-of-the-art QP and VI solvers available from the PETSc library are used on a variety of steady-state 2D and 3D benchmarks, and a comparative study on the scalability between the QP and VI solvers is presented. We then extend the proposed framework to transient problems by simulating the miscible displacement of fluids in a heterogeneous porous medium and illustrate the importance of enforcing maximum principles for these types of coupled problems. Our numerical experiments indicate that VIs are indeed a viable approach for enforcing the maximum principles and the non-negative constraint in a large-scale computing environment. Also provided are Firedrake project files as well as a discussion on the computer implementation to help facilitate readers in understanding the proposed framework.

Comparative Study of Finite Element Methods Using the Time-Accuracy-Size(TAS) Spectrum Analysis

“…Here P p (E) denotes the space of polynomials in d variables of degree less than or equal to p over the element E, and Q p (E) is the space of d-dimensional tensor products of polynomials of degree less than or equal to p. The general form of the weak formulation for equation (15) can be written as follows: Find u h ∈ U h such that…”

“…One of the most popular approaches undertaken is the development of sophisticated finite element packages like FEniCS/Dolfin [4,30], deal.II [3,9], Firedrake [35], LibMesh [25], and MOOSE [21] which provide application scientists the necessary scientific tools to quickly address their specific needs. Alternatively, stand alone finite element computational frameworks built on top of parallel linear algebra libraries like PETSc [7,8] may need to be developed to address specific technical problems such as enforcing maximum principles in subsurface flow and transport modeling [14,15,31] or modeling atmospheric and other geophysical phenomena [13,32], all of which could require field-scale or even global-scale resolutions. As scientific problems grow increasingly complex, the software and algorithms used must be fast, scalable, and efficient across a wide range of hardware architectures and scientific applications, and new algorithms and numerical discretizations may need to be introduced.…”

A Segregated Approach for Modeling the Electrochemistry in the 3-D Microstructure of Li-Ion Batteries and Its Acceleration Using Block Preconditioners

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A performance spectrum for parallel computational frameworks that solve PDEs

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