Stormer-Numerov HDG Methods for Acoustic Waves

We introduce and analyze the first energy-conservative hybridizable discontinuous Galerkin method for the semidiscretization in space of the acoustic wave equation. We prove optimal convergence and superconvergence estimates for the semidiscrete method. We then introduce a two-step fourth-order-in-time Stormer-Numerov discretization and prove energy conservation and convergence estimates for the fully discrete method. In particular, we show that by using polynomial approximations of degree two, convergence of order four is obtained. Numerical experiments verifying that our theoretical orders of convergence are sharp are presented. We also show experiments comparing the method with dissipative methods of the same order.     

A Posteriori Error Estimates for HDG Methods

We present the first a posteriori error analysis of the so-called hybridizable discontinuous Galerkin (HDG) methods for second-order elliptic problems. We show that the error in the flux can be controlled by only two terms. The first term captures the so-called data oscillation. The second solely depends on the difference between the trace of the scalar approximation and the corresponding numerical trace. Numerical experiments verifying the reliability and efficiency of the estimate in two-space dimensions are presented.     

Analysis of HDG Methods for Oseen Equations

We propose a hybridizable discontinuous Galerkin (HDG) method to numerically solve the Oseen equations which can be seen as the linearized version of the incompressible Navier-Stokes equations. We use same polynomial degree to approximate the velocity, its gradient and the pressure. With a special projection and postprocessing, we obtain optimal convergence for the velocity gradient and pressure and superconvergence for the velocity. Numerical results supporting our theoretical results are provided.     

Dispersion Analysis of HDG Methods

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.     

A Comparison of HDG Methods for Stokes Flow

In this paper, we compare hybridizable discontinuous Galerkin (HDG) methods for numerically solving the velocity-pressure-gradient, velocity-pressure-stress, and velocity-pressure-vorticity formulations of Stokes flow. Although they are defined by using different formulations of the Stokes equations, the methods share several common features. First, they use polynomials of degree k for all the components of the approximate solution. Second, they have the same globally coupled variables, namely, the approximate trace of the velocity on the faces and the mean of the pressure on the elements. Third, they give rise to a matrix system of the same size, sparsity structure and similar condition number. As a result, they have the same computational complexity and storage requirement. And fourth, they can provide, by means of an element-by element postprocessing, a new approximation of the velocity which, unlike the original velocity, is divergence-free and H(div)-conforming. We present numerical results showing that each of the approximations provided by these three methods converge with the optimal order of k+1 in L ² for any k≥0. We also display experiments indicating that the postprocessed velocity is a better approximation than the original approximate velocity. It converges with an additional order than the original velocity for the gradient-based HDG, and with the same order for the vorticity-based HDG methods. For the stress-based HDG methods, it seems to converge with an additional order for even polynomial degree approximations. Finally, the numerical results indicate that the method based on the velocity-pressure-gradient formulation provides the best approximations for similar computational complexity.     

Upscaled HDG Methods for Brinkman Equations with High-Contrast Heterogeneous Coefficient

In this paper, we present new upscaled HDG methods for Brinkman equations in the context of high-contrast heterogeneous media. The a priori error estimates are derived in terms of both fine and coarse scale parameters that depend on the high-contrast coefficient weakly. Due to the heterogeneity of the problem, a huge global system will be produced after the numerical discretization of HDG method. Thanks to the upscaled structure of the proposed methods, we are able to reduce the huge global system onto the skeleton of the coarse mesh only while still capturing important fine scale features of this problem. The finite element space over the coarse mesh is irrelevant to the fine scale computation. This feature makes our proposed method very attractive. Several numerical examples are presented to support our theoretical findings.     

A Superconvergent HDG Method for the Maxwell Equations

We present and analyze a new hybridizable discontinuous Galerkin (HDG) method for the steady state Maxwell equations. In order to make the problem well-posed, a condition of divergence is imposed on the electric field. Then a Lagrange multiplier p is introduced, and the problem becomes the solution of a mixed curl–curl formulation of the Maxwell’s problem. We use polynomials of degree \(k+1\), k, k to approximate \({{\varvec{u}}},\nabla \times {{\varvec{u}}}\) and p respectively. In contrast, we only use a non-trivial subspace of polynomials of degree \(k+1\) to approximate the numerical tangential trace of the electric field and polynomials of degree \(k+1\) to approximate the numerical trace of the Lagrange multiplier on the faces. On the simplicial meshes, we show that the convergence rates for \(\varvec{u}\) and \(\nabla \times \varvec{u}\) are independent of the Lagrange multiplier p. If we assume the dual operator of the Maxwell equation on the domain has adequate regularity, we show that the convergence rate for \(\varvec{u}\) is \(O(h^{k+2})\). From the point of view of degrees of freedom of the globally coupled unknown: numerical trace, this HDG method achieves superconvergence for the electric field without postprocessing. Finally, we show that the superconvergence of the HDG method is also derived on general polyhedral elements. Numerical results are given to verify the theoretical analysis.     

Arches and Culture

Technological innovation is the driving force of our civilization. Therefore, we assume all other civilizations would exploit a technological advantage to the same degree that we would. We forget, however, that technology is an aspect of culture, and as with any other aspect of culture, it may be more or less important to any given civilization. The history of the arch is an interesting case in point. The arch is a structural device in architecture that has distinct advantages over post-and-beam construction. People have known how to build the arch and how to use it since the third millennium B.C., but did not use it because its form and meaning did not fit with other dimensions of their respective cultures.     

Arches: a Framework for Modeling Complex Terrains

In this paper, we present a framework for representing complex terrains with such features as overhangs, arches and caves and including different materials such as sand and rocks. Our hybrid model combines a volumetric discrete data structure that stores the different materials and an implicit representation for sculpting and reconstructing the surface of the terrain. Complex scenes can be edited and sculpted interactively with high level tools. We also propose an original rock generation technique that enables us to automatically generate complex rocky sceneries with piles of rocks without any computationally demanding physically-based simulation.     

A high-order HDG method for the Biot’s consolidation model

We propose a novel high-order HDG method for the Biot’s consolidation model in poroelasticity. We present optimal $h$-version error analysis for both the semi-discrete and full-discrete (combined with temporal backward differentiation formula) schemes. Numerical tests are provided to demonstrate the performance of the method.     

A Superconvergent HDG Method for Distributed Control of Convection Diffusion PDEs

We consider a distributed optimal control problem governed by an elliptic convection diffusion PDE, and propose a hybridizable discontinuous Galerkin method to approximate the solution. We use polynomials of degree \(k+1\) to approximate the state and dual state, and polynomials of degree \(k \ge 0\) to approximate their fluxes. Moreover, we use polynomials of degree k to approximate the numerical traces of the state and dual state on the faces, which are the only globally coupled unknowns. We prove optimal a priori error estimates for all variables when \( k \ge 0 \). Furthermore, from the point of view of the number of degrees of freedom of the globally coupled unknowns, this method achieves superconvergence for the state, dual state, and control when \(k\ge 1\). We illustrate our convergence results with numerical experiments.     

An HDG Method with Orthogonal Projections in Facet Integrals

We propose and analyze a new hybridizable discontinuous Galerkin (HDG) method for second-order elliptic problems. Our method is obtained by inserting the \(L^2\)-orthogonal projection onto the approximate space for a numerical trace into all facet integrals in the usual HDG formulation. The orders of convergence for all variables are optimal if we use polynomials of degree \(k+l\), \(k+1\) and k, where k and l are any non-negative integers, to approximate the vector, scalar and trace variables, which implies that our method can achieve superconvergence for the scalar variable without postprocessing. Numerical results are presented to verify the theoretical results.     

To CG or to HDG: A Comparative Study

Hybridization through the border of the elements (hybrid unknowns) combined with a Schur complement procedure (often called static condensation in the context of continuous Galerkin linear elasticity computations) has in various forms been advocated in the mathematical and engineering literature as a means of accomplishing domain decomposition, of obtaining increased accuracy and convergence results, and of algorithm optimization. Recent work on the hybridization of mixed methods, and in particular of the discontinuous Galerkin (DG) method, holds the promise of capitalizing on the three aforementioned properties; in particular, of generating a numerical scheme that is discontinuous in both the primary and flux variables, is locally conservative, and is computationally competitive with traditional continuous Galerkin (CG) approaches. In this paper we present both implementation and optimization strategies for the Hybridizable Discontinuous Galerkin (HDG) method applied to two dimensional elliptic operators. We implement our HDG approach within a spectral/hp element framework so that comparisons can be done between HDG and the traditional CG approach.     

Building problem-solving environments with the Arches framework

The computational problems that scientists face are rapidly escalating in size and scope. Moreover, the computer systems used to solve these problems are becoming significantly more complex than the familiar, well-understood sequential model on their desktops. While it is possible to re-train scientists to use emerging high-performance computing (HPC) models, it is much more effective to provide them with a higher-level programming environment that has been specialized to their particular domain. By fostering interaction between HPC specialists and the domain scientists, problem-solving environments (PSEs) provide a collaborative environment. A PSE environment allows scientists to focus on expressing their computational problem while the PSE and associated tools support mapping that domain-specific problem to a high-performance computing system.This article describes Arches, an object-oriented framework for building domain-specific PSEs. The framework was designed to support a wide range of problem domains and to be extensible to support very different high-performance computing targets. To demonstrate this flexibility, two PSEs have been developed from the Arches framework to solve problem in two different domains and target very different computing platforms. The Coven PSE supports parallel applications that require large-scale parallelism found in cost-effective Beowulf clusters. In contrast, RCADE targets FPGA-based reconfigurable computing and was originally designed to aid NASA Earth scientists studying satellite instrument data.     

A Superconvergent HDG Method for Stokes Flow with Strongly Enforced Symmetry of the Stress Tensor

This work proposes a superconvergent hybridizable discontinuous Galerkin (HDG) method for the approximation of the Cauchy formulation of the Stokes equation using same degree of polynomials for the primal and mixed variables. The novel formulation relies on the well-known Voigt notation to strongly enforce the symmetry of the stress tensor. The proposed strategy introduces several advantages with respect to the existing HDG formulations. First, it remedies the suboptimal behavior experienced by the classical HDG method for formulations involving the symmetric part of the gradient of the primal variable. The optimal convergence of the mixed variable is retrieved and an element-by-element postprocess procedure leads to a superconvergent velocity field, even for low-order approximations. Second, no additional enrichment of the discrete spaces is required and a gain in computational efficiency follows from reducing the quantity of stored information and the size of the local problems. Eventually, the novel formulation naturally imposes physical tractions on the Neumann boundary. Numerical validation of the optimality of the method and its superconvergent properties is performed in 2D and 3D using meshes of different element types.     

A priori and a posteriori error analyses of an HDG method for the Brinkman problem

In this paper we introduce and analyze a hybridizable discontinuous Galerkin (HDG) method for the linear Brinkman model of porous media flow in two and three dimensions and with non-homogeneous Dirichlet boundary conditions. We consider a fully-mixed formulation in which the main unknowns are given by the pseudostress, the velocity and the trace of the velocity, whereas the pressure is easily recovered through a simple postprocessing. We show that the corresponding continuous and discrete schemes are well-posed. In particular, we use the projection-based error analysis in order to derive a priori error estimates. Furthermore, we develop a reliable and efficient residual-based a posteriori error estimator, and propose the associated adaptive algorithm for our HDG approximation. Finally, several numerical results illustrating the performance of the method, confirming the theoretical properties of the estimator and showing the expected behavior of the adaptive refinements are presented.