Hybridizable Discontinuous Galerkin Method for Linear Hyperbolic Integro-Differential Equations

HDG method for linear parabolic integro-Differential equations

An explicit hybridizable discontinuous Galerkin method for the acoustic wave equation

A hybridizable discontinuous Galerkin (HDG) formulation of the linearized incompressible Navier-Stokes equations, known as Oseen equations, is presented. The Cauchy stress formulation is considered and the symmetry of the stress tensor and the mixed variable, namely the scaled strain-rate tensor, is enforced pointwise via Voigt notation. Using equal-order polynomial approximations of degree k for all variables, HDG provides a stable discretization. Moreover, owing to Voigt notation, optimal convergence of order k+1 is obtained for velocity, pressure and strain-rate tensor and a local postprocessing strategy is devised to construct an approximation of the velocity superconverging with order k+2, even for low-order polynomial approximations. A tutorial for the numerical solution of incompressible flow problems using HDG is presented, with special emphasis on the technical details required for its implementation.

A phase-based hybridizable discontinuous Galerkin method for the numerical solution of the Helmholtz equation

A previously presented overset mesh enabled hybridizable discontinuous Galerkin (HDG) finite element method is extended in this work to an isentropic compressible (pseudo-compressible) fluid. This formulation is a first-principles approach and is complementary to the augmented Lagrangian approach that was utilized in the previous HDG incompressible Navier–Stokes formulations which eliminate the global pressure field. This is the first original presentation combining overset meshes, HDG, and fluid flow, specifically isentropic flow for low Mach number applications. Verification of the code implementation of the proposed overset-HDG formulation is performed via the method of manufactured solutions (MMS) on a successively refined overset mesh configuration containing five meshes, and for order k=1,…,4, Lagrange polynomial elements in both two and three dimensions. Optimal order convergence, k + 1, can be observed in all fields for both the two- and three-dimensional simulations, for each mesh. A two-dimensional benchmark problem is also presented to enable code-to-code comparison as a preliminary validation exercise.

We present a systematic and constructive methodology to devise various hybridized discontinuous Galerkin (HDG) methods for linearized shallow water equations. It is shown that using the Rankine--Hugoniot condition to solve the Riemann problem is a natural approach to deriving HDG methods. At the heart of our development is an upwind HDG framework obtained by hybridizing the upwind flux in the standard discontinuous Galerkin (DG) approach. Essentially, the HDG framework is a redesign of the standard DG approach to reducing the number of coupled unknowns. An upwind and three other HDG methods are constructed and analyzed for linearized shallow water systems. Rigorous stability and convergence analysis for both semidiscrete and fully discrete systems are provided. We extend the upwind HDG method to a family of penalty HDG schemes and rigorously analyze their well-posedness, stability, and convergence rates. Numerical results for the linear standing wave and the Kelvin wave for oceanic shallow water systems are presented to verify our theoretical findings.

In this paper, the mathematical formulation for a quadratic optimal control problem governed by a linear hyperbolic integro-differential equation is established. We first show the existence and regularity for the solution of the optimal control problem. The finite element approximation is based on the optimality conditions, which are also derived. Then the a priori error estimates for its finite element approximation are obtained with the optimal convergence order. Furthermore some numerical tests are presented to verify the theoretical results.

We propose an upwind hybrid discontinuous Galerkin (HDG) method for the first-order hyperbolic linear Boltzmann transport equation, featuring a flexible expansion suitable for multiscale scenarios. Within the HDG scheme, primal variables and numerical traces are introduced within and along faces of elements, respectively, interconnected through projection matrices. Given the variables in two stages, the HDG method offers significant flexibility in the selection of spatial orders. The global matrix system in this framework is exclusively constructed from numerical traces, thereby effectively reducing the degrees of freedom (DoFs). Additionally, the matrix system in each discrete direction features a blocked-lower-triangular stencil, enhancing the efficiency of solving hyperbolic equationsthrough an upwind sweep sequence. Based on the proposed method, we perform an asymptotic analysis of the upwind-HDG method in the thick diffusion limit. The result reveals that the correct convergence of the upwind-HDG is closely associated with the properties of the response matrix L. A series of numerical experiments, including comparisons with the even-parity HDG, confirms the accuracy and stability of the upwind-HDG method in managing thick diffusive regimes and multiscale heterogeneous problems.

In this paper, we present a new accelerating procedure in order to speed up the convergence of Newton-type methods. In particular, we derive iterations with a high and optimal order of convergence. This technique can be applied to any iteration with a low order of convergence.
 As expected, the convergence of the proposed methods was remarkably fast. The effectiveness of this technique is illustrated by numerical experiments.

We further investigate the high order positivity-preserving discontinuous Galerkin (DG) methods for linear hyperbolic and radiative transfer equations developed in Yuan et al. (SIAM J Sci Comput 38:A2987–A3019, 2016). The DG methods in Yuan et al. (2016) can maintain positivity and high order accuracy, but they rely both on the scaling limiter in Zhang and Shu (J Comput Phys 229:8918–8934, 2010) and a rotational limiter, the latter may alter cell averages of the unmodulated DG scheme, thereby affecting conservation. Even though a Lax–Wendroff type theorem is proved in Yuan et al. (2016), guaranteeing convergence to weak solutions with correct shock speed when such rotational limiter is applied, it would still be desirable if a conservative DG method without changing the cell averages can be obtained which has both high order accuracy and positivity-preserving capability. In this paper, we develop and analyze such a DG method for both linear hyperbolic equations and radiative transfer equations. In the one-dimensional case, the method uses traditional DG space $$P^k$$ of piecewise polynomials of degree at most k. A key result is proved that the unmodulated DG solver in this case can maintain positivity of the cell average if the inflow boundary value and the source term are both positive, therefore the positivity-preserving framework in Zhang and Shu (2010) can be used to obtain a high order conservative positivity-preserving DG scheme. Unfortunately, in two-dimensions this is no longer the case. We show that the unmodulated DG solver based either on $$P^k$$ or $$Q^k$$ spaces (piecewise kth degree polynomials or piecewise tensor-product kth degree polynomials) could generate negative cell averages. We augment the DG space with additional functions so that the positivity of cell averages from the unmodulated DG solver can be restored, thereby leading to high order conservative positivity-preserving DG scheme based on these augmented DG spaces following the framework in Zhang and Shu (2010). Computational results are provided to demonstrate the good performance of our DG schemes.

\n We build a bridge between the hybrid high-order (HHO) and the hybridizable discontinuous Galerkin (HDG) methods in the setting of a model diffusion problem. First, we briefly recall the construction of HHO methods and derive some new variants. Then, by casting the HHO method in mixed form, we identify the numerical flux so that the HHO method can be compared to HDG methods. In turn, the incorporation of the HHO method into the HDG framework brings up new, efficient choices of the local spaces and a new, subtle construction of the numerical flux ensuring optimal orders of convergence on meshes made of general shape-regular polyhedral elements. Numerical experiments comparing two of these methods are shown.\n

SummaryWe introduce the use of hybridizable discontinuous Galerkin (HDG) finite element methods on overlapping (overset) meshes. Overset mesh methods are advantageous for solving problems on complex geometrical domains. We combine geometric flexibility of overset methods with the advantages of HDG methods: arbitrarily high‐order accuracy, reduced size of the global discrete problem, and the ability to solve elliptic, parabolic, and/or hyperbolic problems with a unified form of discretization. Our approach to developing the ‘overset HDG’ method is to couple the global solution from one mesh to the local solution on the overset mesh. We present numerical examples for steady convection–diffusion and static elasticity problems. The examples demonstrate optimal order convergence in all primal fields for an arbitrary amount of overlap of the underlying meshes. Copyright © 2017 John Wiley & Sons, Ltd.

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This work presents a dispersion analysis of the Hybrid Discontinuous Galerkin (HDG) method. Considering the Helmholtz system, we quantify the discrepancies between the exact and discrete wavenumbers. In particular, we obtain an analytic expansion for the wavenumber error for the lowest order Single Face HDG (SFH) method. The expansion shows that the SFH method exhibits convergence rates of the wavenumber errors comparable to that of the mixed hybrid Raviart–Thomas method. In addition, we observe the same behavior for the higher order cases in numerical experiments.

The local discontinuous Galerkin method for convection-diffusion-fractional anti-diffusion equations