On enforcing maximum principles and achieving element-wise species balance for advection–diffusion–reaction equations under the finite element method

Mudunuru, Maruti K.; Nakshatrala, K. B.

doi:10.1016/j.jcp.2015.09.057

Cited by 27 publications

(30 citation statements)

References 57 publications

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“…However, maximum principles requiring milder regularity conditions on the solution, even for the case when Neumann boundary conditions are prescribed on the boundary, can be found in literature (see [29,30]). If f (x) ≥ 0 in Ω and c p (x) ≥ 0 on the entire ∂Ω then the maximum principle implies that c(x) ≥ 0 in the entire domain, which is the non-negativity of the concentration field.…”

Section: Governing Equations and Associated Non-negative Numerical Mementioning

confidence: 99%

Large-Scale Optimization-Based Non-negative Computational Framework for Diffusion Equations: Parallel Implementation and Performance Studies

2016

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This figure shows the fate of chromium after 180 days using the single-field Galerkin formulation. The white regions indicate the violation of the non-negative constraint. Abstract. It is well-known that the standard Galerkin formulation, which is often the formulation of choice under the finite element method for solving self-adjoint diffusion equations, does not meet maximum principles and the non-negative constraint for anisotropic diffusion equations. Recently, optimization-based methodologies that satisfy maximum principles and the non-negative constraint for steady-state and transient diffusion-type equations have been proposed. To date, these methodologies have been tested only on small-scale academic problems. The purpose of this paper is to systematically study the performance of the non-negative methodology in the context of high performance computing (HPC). PETSc and TAO libraries are, respectively, used for the parallel environment and optimization solvers. For large-scale problems, it is important for computational scientists to understand the computational performance of current algorithms available in these scientific libraries. The numerical experiments are conducted on the state-of-the-art HPC systems, and a single-core performance model is used to better characterize the efficiency of the solvers. Our studies indicate that the proposed non-negative computational framework for diffusion-type equations exhibits excellent strong scaling for real-world large-scale problems.

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Section: Governing Equations and Associated Non-negative Numerical Mementioning

confidence: 99%

Large-Scale Optimization-Based Non-negative Computational Framework for Diffusion Equations: Parallel Implementation and Performance Studies

2016

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“…A generalization of the classical maximum principle that is relevant to this paper is provided in [Mudunuru and Nakshatrala, 2016a]. Specifically, they have extended the classical maximum principle on four fronts: the regularity of the solution is relaxed to C 1 (Ω) ∩ C 0 (Ω), the regularity of the volumetric source f (x) is relaxed to the space of square integrable functions, the boundary can have both Dirichlet and Neumann boundary conditions, and the Neumann boundary conditions are further divided into inflow and outflow (i.e., similar to equations (2.1a)-(2.1b)).…”

Section: Strong Problems (Sp)mentioning

confidence: 99%

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

Chang

Nakshatrala

2017

Computer Methods in Applied Mechanics and Engineering

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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.

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“…The maximum principle is used to obtain physical meaningful numerical solutions (see [Nakshatrala et al, 2015;Mudunuru and Nakshatrala, 2016] and References therein). Herein, it can be utilized to verify the accuracy of numerical solutions by plotting vorticity and checking whether the non-negative maximum and non-positive minimum of the vorticity occur on the boundary.…”

Section: Mathematical Properties: Statements and Derivationsmentioning

confidence: 99%

Mechanics-Based Solution Verification for Porous Media Models

Shabouei

Nakshatrala

2016

Commun. Comput. Phys.

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Abstract. This paper presents a new approach to verify accuracy of computational simulations. We develop mathematical theorems which can serve as robust a posteriori error estimation techniques to identify numerical pollution, check the performance of adaptive meshes, and verify numerical solutions. We demonstrate performance of this methodology on problems from flow thorough porous media. However, one can extend it to other models. We construct mathematical properties such that the solutions to Darcy and Darcy-Brinkman equations satisfy them. The mathematical properties include the total minimum mechanical power, minimum dissipation theorem, reciprocal relation, and maximum principle for the vorticity. All the developed theorems have firm mechanical bases and are independent of numerical methods. So, these can be utilized for solution verification of finite element, finite volume, finite difference, lattice Boltzmann methods and so forth. In particular, we show that, for a given set of boundary conditions, Darcy velocity has the minimum total mechanical power of all the kinematically admissible vector fields. We also show that a similar result holds for Darcy-Brinkman velocity. We then show for a conservative body force, the Darcy and Darcy-Brinkman velocities have the minimum total dissipation among their respective kinematically admissible vector fields. Using numerical examples, we show that the minimum dissipation and total mechanical power theorems can be utilized to identify pollution errors in numerical solutions. The solutions to Darcy and Darcy-Brinkman equations are shown to satisfy a reciprocal relation, which has the potential to identify errors in the numerical implementation of boundary conditions. It is also shown that the vorticity under both steady and transient Darcy-Brinkman equations satisfy maximum principles if the body force is conservative and the permeability is homogeneous and isotropic. A discussion on the nature of vorticity under steady and transient Darcy equations is also presented. Using several numerical examples, we will demonstrate the predictive capabilities of the proposed a posteriori techniques in assessing the accuracy of numerical solutions for a general class of problems, which could involve complex domains and general computational grids.

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On enforcing maximum principles and achieving element-wise species balance for advection–diffusion–reaction equations under the finite element method

Cited by 27 publications

References 57 publications

Large-Scale Optimization-Based Non-negative Computational Framework for Diffusion Equations: Parallel Implementation and Performance Studies

Large-Scale Optimization-Based Non-negative Computational Framework for Diffusion Equations: Parallel Implementation and Performance Studies

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

Mechanics-Based Solution Verification for Porous Media Models

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