Conference Agenda

Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).

Please note that all times are shown in the time zone of the conference. The current conference time is: 8th Dec 2022, 11:46:26pm CET

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Session Overview
Z2b: Discontinuous Galerkin, Darcy, UQ
Wednesday, 14/July/2021:
11:50am - 1:50pm

Session Chair: Stefano Berrone
Virtual location: Zoom 8

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11:50am - 12:10pm

A staggered DG method for the Darcy flow in fractured porous media on polygonal meshes

Lina Zhao1, Dohyun Kim2, Eun-Jae Park2, Eric Chung1

1Department of Mathematics,The Chinese University of Hong Kong, Hong Kong SAR, China; 2School of Mathematics and Computing (Computational Science and Engineering), Yonsei University, Korea, Republic of (South Korea)

In this talk, we present and analyze a staggered DG method for Darcy flows in fractured porous media. Darcy flows in a two-dimensional bulk domain are considered. Fractures are modeled as one-dimensional objects so that flows along the fractures are expressed as lower-dimensional Darcy flows. Then the flows in bulk domain and fractures are coupled by interface conditions. We apply a staggered DG method on polygonal mesh for the bulk domain which is known to be robust to mesh distortion numerically. We carry out a priori error estimates without assumption on the ratio between the mesh size and the edge length. Optimal convergence rates are obtained both theoretically and numerically for a smooth solution. A square-like mesh is considered with small edges to observe the effect of the existence of small edges. We also present numerical experiments with meshes obtained by cutting background meshes by fractures. While these meshes contain small edges and sliver elements, we observe optimal convergence rates for all variables with respect to degrees of freedom.

12:10pm - 12:30pm

An expanded staggered DG method for the heterogeneous diffusion equation

Sanghee Lee, Dohyun Kim, Eun-Jae Park

Yonsei University, Korea, Republic of (South Korea)

In this talk, we present an expanded staggered discontinuous Galerkin method and its a posteriori error estimator for the heterogeneous diffusion problem on general polygonal meshes. In groundwater hydrology, the diffusion coefficient may tend to be zero, which be troublesome in computation. The expanded formulation introduces an additional variable, pressure gradient, to avoid the inversion of the diffusion coefficient matrix. We derive a priori and a posteriori error estimates. A priori error estimates show the optimal convergence rates in L2-norms for the pressure, the flux, and the pressure gradient. Residual-based a posteriori error estimates are conducted and, here, a reliable and efficient error estimator is suggested. Three numerical experiments are considered to observe convergence behavior, robustness to mesh distortion, and the performance of the proposed a posteriori error estimator. The results indicate that our method behaves robustly even on meshes with small edges. Also, the effectivity index of the proposed method is reasonably small even with strong anisotropy compared to that of the standard staggered discontinuous Galerkin method.

12:30pm - 12:50pm

An iterative solver for discontinuous Galerkin methods with Lagrange multipliers

Dongwook Shin

Myongji University, Korea, Republic of (South Korea)

In this talk, we introduce an iterative solver for the high order discontinuous Galerkin method with Lagrange multipliers (DGLM) for elliptic problems. An auxiliary variable is introduced on each edge/face by defining a weak divergence and a weak derivative for the method. In the previous work, stability and error estimates for DGLM were investigated, and a criterion for boundary treatment was proposed to capture sharp boundary layers. Recently, we study an efficient iterative solver. This solver can be localized and parallelized. Several numerical experiments are presented to demonstrate the theoretical results and to test performance of the iterative solver. This solver is comparable to the direct solver if the initial guess is sufficiently close to the solution.

This is joint work with prof. Mi-Young Kim (Inha Univ.) and Dr. Jaemin Shin (KIAPS).

12:50pm - 1:10pm

Polygonal staggered discontinuous Galerkin method for Brinkman problem with applications to Navier-Stokes equations

Lina Zhao1, Eric Chung1, Ming Fai Lam1, Dohyun Kim2, Eun-Jae Park2

1the Chinese University of Hong Kong, Hong Kong S.A.R. (China); 2Yonsei University, Seoul, South Korea

In this talk we present a uniformly robust staggered discontinuous Galerkin method for Brinkman problem on general polygonal meshes. The proposed method is robust in the Stokes and Darcy limits, in addition, hanging nodes can be automatically incorporated into the construction of the method. The proposed method is different from existing staggered DG methods in the sense that we relax the tangential continuity of velocity, which is tailored to yield optimal error estimates for velocity gradient, velocity, and pressure that are independent of the viscosity coefficient. Then we extend the proposed scheme to solve Navier-Stokes equations. The proposed scheme offers salient features in particular it is pressure robust and can handle Navier-Stokes equations with high Reynolds number. Finally, several numerical experiments are provided to validate the theoretical findings and demonstrate the good performances of the proposed method.

1:10pm - 1:30pm


Francesca Bonizzoni1, Fabio Nobile2

1Department of Mathematics, University of Augsburg; 2CSQI - MATHICSE, Ecole Polytechnique Fédérale de Lausanne

The talk deals with the Darcy boundary value problem with lognormal

stochastic permeability endowed with Neumann/Dirichlet homogeneous boundary conditions. We adopt a perturbation approach, namely, we approximate the expected

value of the stochastic solution by the expected value of its Taylor polynomial. The present work focuses on two aspects. First, we derive analytically and study the well-posedness

and regularity (in mixed Hölder spaces) of the recursive problem solved by the expectation of the Taylor polynomial. Secondly, we consider a sparse approximation of the recursion, and we study the convergence of the sparse discretization error.

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