We present a heterogeneous computing strategy for a hybridizable discontinuous Galerkin (HDG) geometric multigrid (GMG) solver. Parallel GMG solvers require a combination of coarse grain and fine grain parallelism is utilized to improve time to solution performance. In this work we focus on fine grain parallelism. We use Intel's second generation Xeon Phi (Knights Landing) to enable acceleration. The GMG method achieves ideal convergence rates of 0.2 or less, for high polynomial orders. A matrix free (assembly free) technique is exploited to save considerable memory usage and increase arithmetic intensity. HDG enables static condensation, and due to the discontinuous nature of the discretization, we developed a matrix-vector multiply routine that does not require any costly synchronizations or barriers. Our algorithm is able to attain 80% of peak bandwidth performance for higher order polynomials. This is possible due to the data locality inherent in the HDG method. Very high performance is realized for high order schemes, due to good arithmetic intensity, which declines as the order is reduced.

The focus of this paper is the analysis of families of hybridizable interior penalty discontinuous Galerkin methods for second order elliptic problems. We derive a priori error estimates in the energy norm that are optimal with respect to the mesh size. Suboptimal L2 norm error estimates are proven. These results are valid in two and three dimensions. Numerical results support our theoretical findings, and we illustrate the computational cost of the method.

We present a new method for simulating incompressible immiscible two-phase flow in porous media. The semi-implicit method decouples the wetting phase pressure and saturation equations. The equations are discretized using a hybridizable discontinuous Galerkin method. The proposed method is of high order, conserves global/local mass balance, and the number of globally coupled degrees of freedom is significantly reduced compared to standard interior penalty discontinuous Galerkin methods. Several numerical examples illustrate the accuracy and robustness of the method. These examples include verification of convergence rates by manufactured solutions, common one-dimensional benchmarks, and realistic discontinuous permeability fields. KEYWORDSdiscontinuous Galerkin, incompressible flow, transport INTRODUCTIONMultiphase flows in porous media are fundamental processes in geophysics. For instance, they characterize enhanced oil recovery, 1 hydrogeology, 2,3 and CO2 sequestration in geological formations. 4 The equations that govern two-phase flow form a coupled system of nonlinear partial differential equations. Numerous techniques have been proposed to resolve the nonlinearity inherent in the system, for instance, implicit-explicit (IMPES), semi-sequential, semi-implicit, and fully implicit methods. 5 The selection of spatial discretization is also a critical decision in the solution process. Accuracy, local conservation, mass balance, and efficiency of implementation are all important features. With respect to the wetting phase pressure equation, incorrect approximations to the phase velocity can cause oscillations and instability when used in the convection-dominated transport equation satisfied by the wetting phase saturation. Compatible discretizations (as defined in the work of Dawson et al 6 ) for flow and transport maintain local and global mass conservation, which provides stability and accuracy in the numerical methods.In this work, we discretize the pressure and saturation equations by the hybridizable discontinuous Galerkin (HDG) method. The HDG method has several interesting properties. In particular, discrete analogs of global conservation for flow and local conservation for transport are satisfied. This postprocessing is available since the normal component of the numerical flux for the HDG method is single valued. 7 Classical discontinuous Galerkin (DG) methods construct the velocity from the pressure, which means that the velocity (and subsequent its H(div) postprocessing) converges sub-optimally. 8 The DG method has received a lot of attention since the 1990s. These DG methods have a number of attractive features, eg, local mass conservation, hp-adaptation, their ability to handle nonconforming meshes, and the fact that they are well Int J Numer Methods Eng. 2018;116:161-177.wileyonlinelibrary.com/journal/nme

We present a performance analysis appropriate for comparing algorithms using different numerical discretizations. By taking into account the total time-to-solution, numerical accuracy with respect to an error norm, and the computation rate, a cost-benefit analysis can be performed to determine which algorithm and discretization are particularly suited for an application. This work extends the performance spectrum model in [16] for interpretation of hardware and algorithmic tradeoffs in numerical PDE simulation. As a proof-of-concept, popular finite element software packages are used to illustrate this analysis for Poisson's equation.

We present a new method for approximating solutions to the incompressible miscible displacement problem in porous media. At the discrete level, the coupled nonlinear system has been split into two linear systems that are solved sequentially. The method is based on a hybridizable discontinuous Galerkin method for the Darcy flow, which produces a mass-conservative flux approximation, and a hybridizable discontinuous Galerkin method for the transport equation. The resulting method is high order accurate. Due to the implicit treatment of the system of partial differential equations, we observe computationally that no slope limiters are needed. Numerical experiments are provided that show that the method converges optimally and is robust for highly heterogeneous porous media in 2D and 3D.