A DG-VEM method for the dissipative wave equation

A novel space-time discretization for the (linear) scalar-valued dissipative wave equation is presented. It is a structured approach, namely, the discretization space is obtained tensorizing the Virtual Element (VE) discretization in space with the Discontinuous Galerkin (DG) method in time. As such, it combines the advantages of both the VE and the DG methods. The proposed scheme is implicit and it is proved to be unconditionally stable and accurate in space and time

A cVEM-DG space-time method for the dissipative wave equation

A novel space-time discretization for the (linear) scalar-valued dissipative wave equation is presented. It is a structured approach, namely, the discretization space is obtained tensorizing the Virtual Element (VE) discretization in space with the Discontinuous Galerkin (DG) method in time. As such, it combines the advantages of both the VE and the DG methods. The proposed scheme is implicit and it is proved to be unconditionally stable and accurate in space and time

A note on optimal spectral bounds for nonoverlapping domain decomposition preconditioners for hp-version discontinuous Galerkin methods

In this article, we consider the derivation of hp-optimal spectral bounds for a class of domain decomposition preconditioners based on the Schwarz framework for discontinuous Galerkin finite element approximations of second-order elliptic partial differential equations. In particular, we improve the bounds derived in our earlier article [P.F. Antonietti and P. Houston, J. Sci. Comput., 46(1):124-149, 2011] in the sense that the resulting bound on the condition number of the preconditioned system is not only explicit with respect to the coarse and fine mesh sizes H and h, respectively, and the fine mesh polynomial degree p, but now also explicit with respect to the polynomial degree q employed for the coarse grid solver. More precisely, we show that the resulting spectral bounds are of order p2H/(qh) for the hp-version of the discontinuous Galerkin method

Dispersion-dissipation analysis of 3D continuous and discontinuous spectral element methods for the elastodynamics equation

In this paper we present a three dimensional dispersion and dissipation analysis for both the semi discrete and the fully discrete approximation of the elastodynamics equation based on the plane wave method. For space discretization we compare different approximation strategies, namely the continuous and the discontinuous spectral element method on both tetrahedral and hexahedral elements. For time discretization we employ a leapfrog time integration scheme. Several numerical results are presented and discussed

The conforming virtual element method for polyharmonic and elastodynamics problems: a review

In this paper, we review recent results on the conforming virtual element approximation of polyharmonic and elastodynamics problems. The structure and the content of this review is motivated by three paradigmatic examples of applications: classical and anisotropic Cahn-Hilliard equation and phase field models for brittle fracture, that are briefly discussed in the first part of the paper. We present and discuss the mathematical details of the conforming virtual element approximation of linear polyharmonic problems, the classical Cahn-Hilliard equation and linear elastodynamics problems.Comment: 30 pages, 7 figures. arXiv admin note: text overlap with arXiv:1912.0712

Discontinuous Galerkin Methods for Fisher-Kolmogorov Equation with Application to α\alpha-Synuclein Spreading in Parkinson's Disease

The spreading of prion proteins is at the basis of brain neurodegeneration. The paper deals with the numerical modelling of the misfolding process of $\alpha$-synuclein in Parkinson's disease. We introduce and analyze a discontinuous Galerkin method for the semi-discrete approximation of the Fisher-Kolmogorov (FK) equation that can be employed to model the process. We employ a discontinuous Galerkin method on polygonal and polyhedral grids (PolyDG) for space discretization, which allows us to accurately simulate the wavefronts typically observed in the prionic spreading. We prove stability and a priori error estimates for the semi-discrete formulation. Next, we use a Crank-Nicolson scheme to advance in time. For the numerical verification of our numerical model, we first consider a manufactured solution, and then we consider a case with wavefront propagation in two-dimensional polygonal grids. Next, we carry out a simulation of $\alpha$-synuclein spreading in a two-dimensional brain slice in the sagittal plane with a polygonal agglomerated grid that takes full advantage of the flexibility of PolyDG approximation. Finally, we present a simulation in a three-dimensional patient-specific brain geometry reconstructed from magnetic resonance images.Comment: arXiv admin note: text overlap with arXiv:2210.0227

Multigrid algorithms for hphp-discontinuous Galerkin discretizations of elliptic problems

Abstract. We present W-cycle multigrid algorithms for the solution of the linear system of equations arising from a wide class of hp-version discontinuous Galerkin discretizations of elliptic problems. Starting from a classical framework in multigrid analysis, we define a smoothing and an approximation property, which are used to prove the uniform convergence of the W-cycle scheme with respect to the granularity of the grid and the number of levels. The dependence of the convergence rate on the polynomial approximation degree p is also tracked, showing that the contraction factor of the scheme deteriorates with increasing p. A discussion on the effects of employing inherited or non-inherited sublevel solvers is also presented. Numerical experiments confirm the theoretical results. Key words. hp-version discontinuous Galerkin, multigrid algorithms, elliptic problem

Level set-fitted polytopal meshes with application to structural topology optimization

We propose a method to modify a polygonal mesh in order to fit the zero-isoline of a level set function by extending a standard body-fitted strategy to a tessellation with arbitrarily-shaped elements. The novel level set-fitted approach, in combination with a Discontinuous Galerkin finite element approximation, provides an ideal setting to model physical problems characterized by embedded or evolving complex geometries, since it allows skipping any mesh post-processing in terms of grid quality. The proposed methodology is firstly assessed on the linear elasticity equation, by verifying the approximation capability of the level set-fitted approach when dealing with configurations with heterogeneous material properties. Successively, we combine the level set-fitted methodology with a minimum compliance topology optimization technique, in order to deliver optimized layouts exhibiting crisp boundaries and reliable mechanical performances. An extensive numerical test campaign confirms the effectiveness of the proposed method