Fully implicit hybrid two-level domain decomposition algorithms for two-phase flows in porous media on 3D unstructured grids

Luo, Li‐Shi; Liu, Lulu; Cai, Xiao‐Chuan; Keyes, David E.

doi:10.1016/j.jcp.2020.109312

Cited by 22 publications

(7 citation statements)

References 58 publications

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“…In this paper, we solve the system of (2.5) and (2.6) simultaneously using a fully coupled approach. The discretization is based on a backward Euler scheme in time and an upwind non-symmetric interior penalty Galerkin (NIPG [16,38]) finite element method in space. This manuscript is for review purposes only.…”

Section: Fully Implicit Discontinuous Galerkin Finite Element Discretizationmentioning

confidence: 99%

“…In this study, we choose to analytically compute J using the chain rule since the exact Jacobian matrix brings added robustness. We refer to [38] for more details for the construction of J.…”

Section: 3mentioning

confidence: 99%

“…Other advantages of DG include the robustness of the method for equations with discontinuous coefficients and the local mass conservation property. We refer to [3,16,38] for more details of the fully implicit DG discretization for modeling two-phase flows in porous media.…”

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confidence: 99%

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Nonlinear Preconditioning Strategies for Two-Phase Flows in Porous Media Discretized by a Fully Implicit Discontinuous Galerkin Method

Luo¹,

Cai²,

Keyes³

2021

SIAM J. Sci. Comput.

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We consider numerical simulation of two-phase flows in porous media using implicit methods. Because of the complex features involving heterogeneous permeability and nonlinear capillary effects the nonlinear algebraic systems arising from the discretization are very difficult to solve. The traditional Newton method suffers from slow convergence in the form of a long stagnation or sometimes does not converge at all. In the paper, we develop some nonlinear preconditioning strategies for the system of two-phase flows discretized by a fully implicit discontinuous Galerkin method. The preconditioners identify and approximately eliminate the local high nonlinearities that cause the Newton method to take small updates. Specifically, we propose two elimination strategies, one is based on exploring the unbalanced nonlinearities of the pressure and the saturation fields, and the other is based on identifying certain elements of the finite element space that have much higher nonlinearities than the rest of the elements. We compare the performance and robustness of the proposed algorithms with an existing single-field elimination approach and the classical inexact Newton method with respect to some physical and numerical parameters. Experiments on three-dimensional porous media applications show that the proposed algorithms are superior to the other methods in terms of the robustness and parallel efficiency.

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Section: Fully Implicit Discontinuous Galerkin Finite Element Discretizationmentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

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Nonlinear Preconditioning Strategies for Two-Phase Flows in Porous Media Discretized by a Fully Implicit Discontinuous Galerkin Method

Luo¹,

Cai²,

Keyes³

2021

SIAM J. Sci. Comput.

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“…The effect is more pronounced for larger scale problems. Borrowing a term from numerical methods for partial differential equations, [21][22][23][24][25] we refer to method (10) as the domain decomposed PCA (DDPCA) in the rest of the paper. The method avoids cross pollution from unrelated parts of the data…”

Section: Summation Pollution Of the Classical Pcamentioning

confidence: 99%

Summation pollution of principal component analysis and an improved algorithm for location sensitive data

Cai

2021

Numerical Linear Algebra App

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Principal component analysis (PCA) is widely used for dimensionality reduction and unsupervised learning. The reconstruction error is sometimes large even when a large number of eigenmode is used. In this paper, we show that this unexpected error source is the pollution effect of a summation operation in the objective function of the PCA algorithm. The summation operator brings together unrelated parts of the data into the same optimization and the result is the reduction of the accuracy of the overall algorithm. We introduce a domain decomposed PCA that improves the accuracy, and surprisingly also increases the parallelism of the algorithm. To demonstrate the accuracy and parallel efficiency of the proposed algorithm, we consider three applications including a face recognition problem, a brain tumor detection problem using two-and three-dimensional MRI images.

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“…ILU( p ) is a fundamental building block of many preconditioning techniques, such as domain decomposition methods in Kong and Cai (2016) and Luo et al (2020), for solving linear system of equations, and is also one of the most difficult components to be parallelized on a GPU because it is originally designed for purely sequential computers. In this paper, an inexact ILU( p ) preconditioner in the point-block form is investigated for a GPU.…”

Section: Some Concluding Remarksmentioning

confidence: 99%

Point-block incomplete LU preconditioning with asynchronous iterations on GPU for multiphysics problems

Cai

2020

The International Journal of High Performance Computing Applica

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Point-block matrices arise naturally in multiphysics problems when all variables associated with a mesh point are ordered together, and are different from the general block matrices since the sizes of the blocks are so small one can often invert some of the diagonal blocks explicitly. Motivated by the recent works of Chow and Patel and Chow et al., we propose an efficient incomplete LU (ILU) preconditioner for point-block matrices targeting applications on GPU. The construction of the preconditioner involves two critical steps: (1) the initial guessing of values for the lower and upper triangular matrices; and (2) several sweeps of asynchronous updating of the triangular matrices. Three representative problems are studied to show the advantage of the proposed point-block approach over the standard point-wise approach in terms of the number of GMRES iterations and also the total compute time. Moreover, we compare the proposed algorithm with the level-scheduling based parallel algorithm employed in NVIDIA’s cuSPARSE library as well as the serial method implemented in Intel MKL library, and the experiments show that a 2×–5× speedup can be achieved over the block-based ILU( p) factorizations from the cuSPARSE library.

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Fully implicit hybrid two-level domain decomposition algorithms for two-phase flows in porous media on 3D unstructured grids

Cited by 22 publications

References 58 publications

Nonlinear Preconditioning Strategies for Two-Phase Flows in Porous Media Discretized by a Fully Implicit Discontinuous Galerkin Method

Nonlinear Preconditioning Strategies for Two-Phase Flows in Porous Media Discretized by a Fully Implicit Discontinuous Galerkin Method

Summation pollution of principal component analysis and an improved algorithm for location sensitive data

Point-block incomplete LU preconditioning with asynchronous iterations on GPU for multiphysics problems

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