Enforcing the non‐negativity constraint and maximum principles for diffusion with decay on general computational grids

Nagarajan, Harsha; Nakshatrala, K. B.

doi:10.1002/fld.2389

Cited by 29 publications

(68 citation statements)

References 61 publications

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“…Since this solution presents maxima and minima in the x direction on the lines x = 0.25 and x = 0.75, the shock detector is activated and AV is added. The problem is solved in triangular meshes of N h × N h (×2) elements with size N h = 12, 24,48,96,192. Even though the presence of the AV increases the error in L 2 (Ω) norm (as expected), the convergence of order 2 is maintained, as it can be appreciated in Fig.…”

Section: Convergence Of a Smooth Solutionmentioning

confidence: 99%

“…These methods are implicit in sense, and usually based on the addition of AV to the problem at hand; they are traditionally called shock (or discontinuity)-capturing techniques, even though we favour the notation nonlinear stabilization. Some approaches to prove a DMP using piecewise higher order polynomials have been done [24,25,20,29,28,31,32] but only the Poisson problem has been proved to enjoy the DMP and only on certain one-dimensional (1D) meshes [29] and on very restrictive quadratic and cubic two dimensional meshes [22,16]. When it comes to discontinuous methods, most of the shock capturing techniques are based on the concept of slope limiter, proposed by Cockburn and Shu for conservation laws [11,10] and latter adapted to the convection-dominated convection-diffusion problem [12].…”

Section: Introductionmentioning

confidence: 99%

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On discrete maximum principles for discontinuous Galerkin methods

Badia

Hierro

2015

Computer Methods in Applied Mechanics and Engineering

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The aim of this work is to propose a monotonicity-preserving method for discontinuous Galerkin (dG) approximations of convection-diffusion problems. To do so, a novel definition of discrete maximum principle (DMP) is proposed using the discrete variational setting of the problem, and we show that the fulfilment of this DMP implies that the minimum/maximum (depending on the sign of the forcing term) is on the boundary for multidimensional problems. Then, an artificial viscosity (AV) technique is designed for convection-dominant problems that satisfies the above mentioned DMP. The noncomplete stabilized interior penalty dG method is proved to fulfil the DMP property for the one-dimensional linear case when adding such AV with certain parameters. The benchmarks for the constant values to satisfy the DMP are calculated and tested in the numerical experiments section. Finally, the method is applied to different test problems in one and two dimensions to show its performance.

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Section: Convergence Of a Smooth Solutionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On discrete maximum principles for discontinuous Galerkin methods

Badia

Hierro

2015

Computer Methods in Applied Mechanics and Engineering

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“…Since neither the finite difference nor finite volume methods are based on weak formulations, these mentioned works cannot be trivially extended to the finite element method. (e) Optimization-based techniques at the discrete level : Several studies over the years [Nakshatrala and Valocchi, 2009;Nagarajan and Nakshatrala, 2011;Nakshatrala et al, 2013Nakshatrala et al, , 2016 have focused on the development of optimization-based methodologies that enforce the maximum principle and the non-negative constraint for diffusion problems. An optimization-based methodology based on the work of the aforementioned studies has been applied to enforce maximum principles advection-diffusion equations [Mudunuru and Nakshatrala, 2016a].…”

Section: Introductionmentioning

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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“…In References [23,22], optimization-based techniques have been employed to meet these constraints under both mixed and single-field finite element formulations but have restricted their studies to low-order finite elements.…”

Section: Introductionmentioning

confidence: 99%

“…Also, earlier numerical works on discrete maximum principles concentrated on single-field formulations [22], and mixed formulations based on the variational multiscale formalism or lowest-order Raviart-Thomas spaces [23]. Herein, we shall also investigate the performance of mixed formulations based on the least-squares formalism.…”

Section: Introductionmentioning

confidence: 99%

On the performance of high‐order finite elements with respect to maximum principles and the nonnegative constraint for diffusion‐type equations

Payette

Nakshatrala

Reddy

2012

Numerical Meth Engineering

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Abstract. The main aim of this paper is to document the performance of p-refinement with respect to maximum principles and the non-negative constraint. The model problem is (steadystate) anisotropic diffusion with decay (which is a second-order elliptic partial differential equation).We considered the standard single-field formulation (which is based on the Galerkin formalism) and two least-squares-based mixed formulations. We have employed non-uniform Lagrange polynomials for altering the polynomial order in each element, and we have used p = 1, · · · , 10. It will be shown that the violation of the non-negative constraint will not vanish with p-refinement for anisotropic diffusion. We shall illustrate the performance of p-refinement using several representative problems.The intended outcome of the paper is twofold. Firstly, this study will caution the users of high-order approximations about its performance with respect to maximum principles and the non-negative constraint. Secondly, this study will help researchers to develop new methodologies for enforcing maximum principles and the non-negative constraint under high-order approximations.

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Enforcing the non‐negativity constraint and maximum principles for diffusion with decay on general computational grids

Cited by 29 publications

References 61 publications

On discrete maximum principles for discontinuous Galerkin methods

On discrete maximum principles for discontinuous Galerkin methods

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

On the performance of high‐order finite elements with respect to maximum principles and the nonnegative constraint for diffusion‐type equations

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