Two new approaches for solving elliptic obstacle problems using discontinuous Galerkin methods

Gaddam, Sharat; Gudi, Thirupathi; Porwal, Kamana

doi:10.1007/s10543-021-00869-w

Cited by 6 publications

(2 citation statements)

References 49 publications

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“…The unilateral obstacle problem has many applications, such as lubrication phenomena, fluid flow in porous media and option pricing, etc. [19,20]. In this paper, we propose a new ADMM [7,21] with an optimal parameter for the unilateral obstacle problem [22][23][24][25][26][27][28][29][30][31], which is based on the extreme eigenvalues of the matrix using the finite difference method (FDM) [6,22,26,32].…”

Section: Introductionmentioning

confidence: 99%

An Optimal ADMM for Unilateral Obstacle Problems

Zhang,

Cui,

Xiong

et al. 2024

Mathematics

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We propose a new alternating direction method of multipliers (ADMM) with an optimal parameter for the unilateral obstacle problem. We first use the five-point difference scheme to discretize the problem. Then, we present an augmented Lagrangian by introducing an auxiliary unknown, and an ADMM is applied to the corresponding saddle-point problem. Through eliminating the primal and auxiliary unknowns, a pure dual algorithm is then used. The convergence of the proposed method is analyzed, and a simple strategy is presented for selecting the optimal parameter, with the largest and smallest eigenvalues of the iterative matrix. Several numerical experiments confirm the theoretical findings of this study.

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Section: Introductionmentioning

confidence: 99%

An Optimal ADMM for Unilateral Obstacle Problems

Zhang,

Cui,

Xiong

et al. 2024

Mathematics

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“…DG methods are also widely used to solve variational inequalities. We refer to [57,58,20,29] and [32,33,6,61,7,34] respectively, for a priori and a posteriori analysis of DG methods for variatonal inequalities of the first kind. The articles [35,51] discuss the convergence analysis of DG methods over uniform mesh and adaptive mesh based on a posteriori error estimator for variational inequalities of the second kind.…”

Section: Introductionmentioning

confidence: 99%

Unified Analysis of Discontinuous Galerkin Methods for Frictional Contact Problem with normal compliance

Porwal¹,

Tanvi²

2021

Preprint

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In this article, a reliable and efficient a posteriori error estimator of residual type is derived for a class of discontinuous Galerkin methods for the frictional contact problem with reduced normal compliance which is modeled as a quasi-variational inequality. We further derive a priori error estimates in the energy norm under the minimal regularity assumption on the exact solution. The convergence behavior of error over uniform mesh and the performance of error estimator are illustrated by the numerical results.

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Pointwise adaptive non-conforming finite element method for the obstacle problem

Porwal,

Singla

2024

Comp. Appl. Math.

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Two new approaches for solving elliptic obstacle problems using discontinuous Galerkin methods

Abstract: The main aim of this article is to present two new ways to solve the elliptic obstacle problem by using discontinuous Galerkin finite element methods.

Cited by 6 publications

References 49 publications

An Optimal ADMM for Unilateral Obstacle Problems

An Optimal ADMM for Unilateral Obstacle Problems

Unified Analysis of Discontinuous Galerkin Methods for Frictional Contact Problem with normal compliance

Pointwise adaptive non-conforming finite element method for the obstacle problem

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