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

Payette, Gregory S.; Nakshatrala, K. B.; Reddy, J. N.

doi:10.1002/nme.4291

Cited by 12 publications

(16 citation statements)

References 33 publications

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“…Several studies over the years have focused on the development of methodologies that enforce the DMP and ensure non-negative solutions [32,33,35,36]. However, these studies did not address how these methods can be used for realistic large-scale subsurface problems that have millions of grid nodes.…”

Section: Introductionmentioning

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: Introductionmentioning

confidence: 99%

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

2016

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“…This is due to the fact that higher-order shape functions change their sign within an element [66]. Figure 23 provides a pictorial description of the problem with appropriate initial and boundary conditions.…”

Section: Transport-controlled Bimolecular Chemical Reactionsmentioning

confidence: 99%

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

Mudunuru

Nakshatrala

2016

Journal of Computational Physics

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These figures show the fate of the product in a transient transport-controlled bimolecular reaction under vortex-stirred mixing. The left figure is obtained using a popular numerical formulation, which violates the non-negative constraint. The right figure is based on the proposed computational framework. These figures clearly illustrate the main contribution of this paper: The proposed computational framework produces physically meaningful results for advective-diffusive-reactive systems, which is not the case with many popular formulations. 2015 Computational & Applied Mechanics LaboratoryOn enforcing maximum principles and achieving element-wise species balance for advection-diffusion-reaction equations under the finite element method M. K. Mudunuru and K. B. NakshatralaDepartment of Civil and Environmental Engineering, University of Houston.Abstract. We present a robust computational framework for advective-diffusive-reactive systems that satisfies maximum principles, the non-negative constraint, and element-wise species balance property. The proposed methodology is valid on general computational grids, can handle heterogeneous anisotropic media, and provides accurate numerical solutions even for very high Péclet numbers. The significant contribution of this paper is to incorporate advection (which makes the spatial part of the differential operator non-self-adjoint) into the non-negative computational framework, and overcome numerical challenges associated with advection. We employ low-order mixed finite element formulations based on least-squares formalism, and enforce explicit constraints on the discrete problem to meet the desired properties. The resulting constrained discrete problem belongs to convex quadratic programming for which a unique solution exists. Maximum principles and the non-negative constraint give rise to bound constraints while element-wise species balance gives rise to equality constraints. The resulting convex quadratic programming problems are solved using an interior-point algorithm. Several numerical results pertaining to advection-dominated problems are presented to illustrate the robustness, convergence, and the overall performance of the proposed computational framework.

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“…Prior works on non-negative formulations. The prior non-negative formulations can be broadly classified into the following five categories: (a) Reporting the violations: In [Payette et al, 2012], several cases of violations of the maximum principle and the non-negative constraint have been showcased for different anisotropic diffusivity tensors. This paper also demonstrates that h-and p-refinements do not eliminate these violations.…”

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

Cited by 12 publications

References 33 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

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

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

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