A robust discontinuous Galerkin scheme on anisotropic meshes

Discontinuous Galerkin (DG) methods are extensions of the usual Galerkin finite element methods. Although there are vast amount of studies on DG methods, most of them have assumed shape-regularity conditions on meshes for both theoretical error analysis and practical computations. In this paper, we present a new symmetric interior penalty DG scheme with a modified penalty term. We show that, without imposing the shape-regularity condition on the meshes, the new DG scheme inherits all of the good properties of standard DG methods, and is thus robust on anisotropic meshes. Numerical experiments confirm the theoretical error estimates obtained.

Stable Filtering Procedures for Nodal Discontinuous Galerkin Methods

We prove that the most common filtering procedure for nodal discontinuous Galerkin (DG) methods is stable. The proof exploits that the DG approximation is constructed from polynomial basis functions and that integrals are approximated with high-order accurate Legendre–Gauss–Lobatto quadrature. The theoretical discussion re-contextualizes stable filtering results for finite difference methods into the DG setting. Numerical tests verify and validate the underlying theoretical results.

DG structure on the length 4 big from small construction

The big from small construction was introduced by Kustin and Miller in [A. Kustin and M. Miller, Constructing big Gorenstein ideals from small ones, J. Algebra 85 (1983) 303–322] and can be used to construct resolutions of tightly double linked Gorenstein ideals. In this paper, we expand on the DG-algebra techniques introduced in [A. Kustin, Use DG methods to build a matrix factorization, preprint (2019), arXiv:1905.11435 ] and construct a DG [Formula: see text]-algebra structure on the length [Formula: see text] big from small construction. The techniques employed involve the construction of a morphism from a Tate-like complex to an acyclic DG [Formula: see text]-algebra exhibiting Poincaré duality. This induces homomorphisms which, after suitable modifications, satisfy a list of identities that end up perfectly encapsulating the required associativity and DG axioms of the desired product structure for the big from small construction.

A Novel Robust Strategy for Discontinuous Galerkin Methods in Computational Fluid Mechanics: Why? When? What? Where?

In this paper we will review a recent emerging paradigm shift in the construction and analysis of high order Discontinuous Galerkin methods applied to approximate solutions of hyperbolic or mixed hyperbolic-parabolic partial differential equations (PDEs) in computational physics. There is a long history using DG methods to approximate the solution of partial differential equations in computational physics with successful applications in linear wave propagation, like those governed by Maxwell’s equations, incompressible and compressible fluid and plasma dynamics governed by the Navier-Stokes and the Magnetohydrodynamics equations, or as a solver for ordinary differential equations (ODEs), e.g., in structural mechanics. The DG method amalgamates ideas from several existing methods such as the Finite Element Galerkin method (FEM) and the Finite Volume method (FVM) and is specifically applied to problems with advection dominated properties, such as fast moving fluids or wave propagation. In the numerics community, DG methods are infamous for being computationally complex and, due to their high order nature, as having issues with robustness, i.e., these methods are sometimes prone to crashing easily. In this article we will focus on efficient nodal versions of the DG scheme and present recent ideas to restore its robustness, its connections to and influence by other sectors of the numerical community, such as the finite difference community, and further discuss this young, but rapidly developing research topic by highlighting the main contributions and a closing discussion about possible next lines of research.

Deep Domain-Adversarial Image Generation for Domain Generalisation

Machine learning models typically suffer from the domain shift problem when trained on a source dataset and evaluated on a target dataset of different distribution. To overcome this problem, domain generalisation (DG) methods aim to leverage data from multiple source domains so that a trained model can generalise to unseen domains. In this paper, we propose a novel DG approach based on Deep Domain-Adversarial Image Generation (DDAIG). Specifically, DDAIG consists of three components, namely a label classifier, a domain classifier and a domain transformation network (DoTNet). The goal for DoTNet is to map the source training data to unseen domains. This is achieved by having a learning objective formulated to ensure that the generated data can be correctly classified by the label classifier while fooling the domain classifier. By augmenting the source training data with the generated unseen domain data, we can make the label classifier more robust to unknown domain changes. Extensive experiments on four DG datasets demonstrate the effectiveness of our approach.