Analysis of a space--time hybridizable discontinuous Galerkin method for the advection--diffusion problem on time-dependent domains

This paper presents the first analysis of a space--time hybridizable discontinuous Galerkin method for the advection--diffusion problem on time-dependent domains. The analysis is based on non-standard local trace and inverse inequalities that are anisotropic in the spatial and time steps. We prove well-posedness of the discrete problem and provide a priori error estimates in a mesh-dependent norm. Convergence theory is validated by a numerical example solving the advection--diffusion problem on a time-dependent domain for approximations of various polynomial degree

An embedded--hybridized discontinuous Galerkin method for the coupled Stokes--Darcy system

We introduce an embedded--hybridized discontinuous Galerkin (EDG--HDG) method for the coupled Stokes--Darcy system. This EDG--HDG method is a pointwise mass-conserving discretization resulting in a divergence-conforming velocity field on the whole domain. In the proposed scheme, coupling between the Stokes and Darcy domains is achieved naturally through the EDG--HDG facet variables. \emph{A priori} error analysis shows optimal convergence rates, and that the velocity error does not depend on the pressure. The error analysis is verified through numerical examples on unstructured grids for different orders of polynomial approximation

Continuous and Discontinuous Finite Element Methods for Coupled Surface-Subsurface Flow and Transport Problems

A weak formulation of the coupled problem of flow and transport is discretized and analyzed numerically. The flow problem is characterized by the Navier-Stokes (or Stokes) equations coupled by Darcy equations. The velocity field is obtained by couplings of finite element and discontinuous Galerkin methods. The concentration equation is solved by an improved discontinuous Galerkin method. Convergence of the schemes is obtained. Numerical examples show the robustness of the method for heterogeneous and fractured porous media

Discontinuous Galerkin Approximations for Computing Electromagnetic Bloch Modes in Photonic Crystals

We analyze discontinuous Galerkin finite element discretizations of the Maxwell equations with periodic coefficients. These equations are used to model the behavior of light in photonic crystals, which are materials containing a spatially periodic variation of the refractive index commensurate with the wavelength of light. Depending on the geometry, material properties and lattice structure these materials exhibit a photonic band gap in which light of certain frequencies is completely prohibited inside the photonic crystal. By Bloch/Floquet theory, this problem is equivalent to a modified Maxwell eigenvalue problem with periodic boundary conditions, which is discretized with a mixed discontinuous Galerkin (DG) formulation using modified Nédélec basis functions. We also investigate an alternative primal DG interior penalty formulation and compare this method with the mixed DG formulation. To guarantee the non-pollution of the numerical spectrum, we prove a discrete compactness property for the corresponding DG space. The convergence rate of the numerical eigenvalues is twice the minimum of the order of the polynomial basis functions and the regularity of the solution of the Maxwell equations.We present both 2D and 3D numerical examples to verify the convergence rate of the mixed DG method and demonstrate its application to computing the band structure of photonic crystals