Abstract : Discontinuous Galerkin methods are widely employed in computational science and engineering fields, as they offer accurate and efficient simulations. In particular, discontinuous Galerkin methods offer appealing features including high-order approximation, hp adaptivity, and local mass conservations, which are particularly important for practical applications. The development, analysis, and applications of discontinuous Galerkin methods have stimulated significant research. The aim of this mini-symposium is to gather experts as well as junior researchers in the field to introduce recent achievements on discontinuous Galerkin methods and the related applications, as well as promote exchanges.
[03246] Bound preserving DG methods for multi-species flow with chemical reactions
Format : Talk at Waseda University
Author(s) :
Jie Du (Tsinghua University)
Abstract : For multispecies chemical reactive flows, the solutions have some physical bounds. The mass fraction does not satisfy a maximum principle and hence it is not easy to preserve the upper bound. Also, most of the bound-preserving techniques available are based on Euler forward method. For problems with stiff source, the time step will be significantly limited. In this work, we will construct third order conservative bound-preserving DG methods to overcome all these difficulties.
[03960] Staggered discontinuous Galerkin methods for the Stokes problem on rectangular grids
Format : Talk at Waseda University
Author(s) :
Hyea Hyun Kim (Kyung Hee University)
Thien Binh Nguyen (Vietnamese-German University)
Gung-Min Gie (University of Louisville)
Chang-Yeol Jung (UNIST)
Abstract : A staggered DG (Discontinous Galerkin) method, that was originally developed on triangular meshes, is
extended to rectangular grids for the second order elliptic problems in the first, second, and fourth authors' previous work. On the rectangular grids, the higher order polynomials in higher dimensions can be easily formed without the need for meshing the physical domain. In this talk, we present the extension of our previous staggered DG method to the Stokes system and provide the optimal error estimate for the given polynomial order. Compared to the triangle based DG methods, we also obtain a better inf-sup stability result for the rectangular based DG methods. Numerical results are presented to confirm our optimal error estimate results.
[02857] A mass conservative scheme for the coupled flow and transport
Format : Talk at Waseda University
Author(s) :
Lina Zhao (City University of Hong Kong)
Shuyu Sun (KAUST)
Abstract : In this talk, I will present a mass conservative scheme for the coupled Brinkman flow and transport, where the flow equations are discretized using staggered DG method and mixed FEM. As such, the interface conditions are naturally incorporated into the formulation. Then the transport equation is discretized using unwinding staggered DG methods. The optimal convergence error estimates for all the variables are carried out. Several numerical experiments are carried out to demonstrate the performance.
[03078] hp-Multigrid preconditioner for a divergence-conforming HDG scheme for the incompressible flow problems
Format : Online Talk on Zoom
Author(s) :
Guosheng Fu (University of Notre Dame)
Wenzheng Kuang (University of Notre Dame)
Abstract : In this study, we present an hp-multigrid preconditioner for a divergence-conforming HDG scheme for the generalized Stokes and the Navier-Stokes equations using an augmented Lagrangian formulation. Our method relies on conforming simplicial meshes in two- and three-dimensions. The hp-multigrid algorithm is a multiplicative auxiliary space preconditioner that employs the lowest-order space as the auxiliary space, and we developed a geometric multigrid method as the auxiliary space solver. For the generalized Stokes problem, the crucial ingredient of the geometric multigrid method is the equivalence between the condensed lowest-order divergence-conforming HDG scheme and a Crouzeix-Raviart discretization with a pressure-robust treatment as introduced in Linke and Merdon (Comput. Methods Appl. Mech. Engrg., 311 (2016)), which allows for the direct application of geometric multigrid theory on the Crouzeix-Raviart discretization. The numerical experiments demonstrate the robustness of the proposed $hp$-multigrid preconditioner with respect to mesh size and augmented Lagrangian parameter, with iteration counts insensitivity to polynomial order increase. Inspired by the works by Benzi & Olshanskii (SIAM J. Sci. Comput., 28(6) (2006)) and Farrell et al. (SIAM J. Sci. Comput., 41(5) (2019)), we further test the proposed preconditioner on the divergence-conforming HDG scheme for the Navier-Stokes equations. Numerical experiments show a mild increase in the iteration counts of the preconditioned GMRes solver with the rise in Reynolds number up to 10^3.
[04124] Numerical Modelling of the Brain Poromechanics by High-Order Discontinuous Galerkin Methods
Format : Online Talk on Zoom
Author(s) :
Paola Francesca Antonietti (Politecnico di Milano)
Mattia Corti (Politecnico di Milano)
Luca Dede' (Politecnico di Milano)
Alfio Quarteroni (Politecnico di Milano)
Abstract : In this talk we introduce and analyze a discontinuous Galerkin method for the numerical modelling of the equations of Multiple-Network Poroelastic Theory (MPET) in the dynamic formulation. The MPET model can comprehensively describe functional changes in the brain considering multiple scales of fluids. Concerning the spatial discretization, we employ a high-order discontinuous Galerkin method on polygonal and polyhedral grids and we derive stability and a priori error estimates. The temporal discretization is based on a coupling between a Newmark β-method for the momentum equation and a θ-method for the pressure equations. We present verification numerical results and perform a convergence analysis using an agglomerated mesh of a geometry of a brain slice. We also present a simulation in a three-dimensional patient-specific brain reconstructed from magnetic resonance images. The model presented in this paper can be regarded as a preliminary attempt to model perfusion in the brain.
[03151] Cell-average based Neural Network method for time dependent PDEs
Format : Talk at Waseda University
Author(s) :
Changxin Qiu (Ningbo University)
Jue Yan (Iowa State University)
Abstract : Motivated by finite volume scheme, a cell-average based neural network method is proposed. The method is based on the integral or weak formulation of partial differential equations. Offline supervised training is carried out to obtain the optimal network parameter set, which uniquely identifies one finite volume like neural network method. Once well trained, the network method is implemented as a finite volume scheme and can adapt large time step size for solution evolution.
[04606] A non-overlapping Schwarz algorithm for the HDG method
Format : Talk at Waseda University
Author(s) :
Issei Oikawa (University of Tsukuba)
Abstract : This talk is concerned with a non-overlapping Schwarz algorithm for the hybridizable discontinuous Galerkin (HDG) method for the steady-state diffusion problem. We present several iterative algorithms based on the non-overlapping Schwarz domain decomposition method and their numerical results.
[03898] Adaptive methods for fully nonlinear PDE
Format : Online Talk on Zoom
Author(s) :
Iain Smears (University College London)
Abstract : Hamilton--Jacobi--Bellman and Isaacs equations are important classes of fully nonlinear PDE with applications from stochastic optimal control and two player stochastic differential games. In this talk, we present our recent proof of the convergence of a broad family of adaptive nonconforming DG and $C^0$-interior penalty methods for the class of these equations that satisfy the Cordes condition in two or three space dimensions. The adaptive mesh refinement is driven by reliable and efficient a posteriori error estimators, and convergence is proven in $H^2$-type norms without higher regularity assumptions of the solution. A foundational ingredient in the proof of convergence is the concept of the limit space used to describe the limiting behaviour of the finite element spaces under the adaptive mesh refinement algorithm. We develop a novel approach to the construction and analysis of these nonstandard function spaces via intrinsic characterizations in terms of the distributional derivatives of functions of bounded variation. We provide a detailed theory for the limit spaces, and also some original auxiliary function spaces, that resolves some foundational challenges and that is of independent interest to adaptive nonconforming methods for more general problems. These include Poincare and trace inequalities, a proof of the density of functions with nonvanishing jumps on only finitely many faces of the limit skeleton, symmetry of the Hessians, approximation results by finite element functions and weak convergence results.
