Discontinuous Galerkin methods for convection-diffusion equations and applications in petroleum engineering

This dissertation contains research in discontinuous Galerkin (DG) methods applying to convection-diffusion equations. It contains both theoretical analysis and applications. Initially, we develop a conservative local discontinuous Galerkin (LDG) method for the coupled system of compressible miscible displacement problem in two space dimensions. The main difficulty is how to deal with the discontinuity of approximations of velocity, u, in the convection term across the cell interfaces. To overcome the problems, we apply the idea of LDG with IMEX time marching using the diffusion term to control the convection term. Optimal error estimates in Linfinity(0, T; L2) norm for the solution and the auxiliary variables will be derived. Then, high-order bound-preserving (BP) discontinuous Galerkin (DG) methods for the coupled system of compressible miscible displacements on triangular meshes will be developed. There are three main difficulties to make the concentration of each component between 0 and 1. Firstly, the concentration of each component did not satisfy a maximum-principle. Secondly, the first-order numerical flux was difficult to construct. Thirdly, the classical slope limiter could not be applied to the concentration of each component. To conquer these three obstacles, we first construct special techniques to preserve two bounds without using the maximum-principle-preserving technique. The time derivative of the pressure was treated as a source of the concentration equation. Next, we apply the flux limiter to obtain high-order accuracy using the second-order flux as the lower order one instead of using the first-order flux. Finally, L2-projection of the porosity and constructed special limiters that are suitable for multi-component fluid mixtures were used. Lastly, a new LDG method for convection-diffusion equations on overlapping mesh introduced in [J. Du, Y. Yang and E. Chung, Stability analysis and error estimates of local discontinuous Galerkin method for convection-diffusion equations on overlapping meshes, BIT Numerical Mathematics (2019)] showed that the convergence rates cannot be improved if the dual mesh is constructed by using the midpoint of the primitive mesh. They provided several ways to gain optimal convergence rates but the reason for accuracy degeneration is still unclear. We will use Fourier analysis to analyze the scheme for linear parabolic equations with periodic boundary conditions in one space dimension. To investigate the reason for the accuracy degeneration, we explicitly write out the error between the numerical and exact solutions. Moreover, some superconvergence points that may depend on the perturbation constant in the construction of the dual mesh were also found out

Spectral/hp element methods: recent developments, applications, and perspectives

The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials, modified to accommodate C0-continuous expansions. Computationally and theoretically, by increasing the polynomial order p, high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use the spectral/hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order to use the spectral/hp element method in more complex science and engineering applications are discussed

The parallel finite element system M++ with integrated multilevel preconditioning and multilevel Monte Carlo methods

We present a parallel data structure for the discretization of partial differential equations which is based on distributed point objects and which enables the flexible, transparent, and efficient realization of conforming, nonconforming, and mixed finite elements. This concepts is realized for elliptic, parabolic and hyperbolic model problems, and sample applications are provided by a tutorial complementing a lecture on scientific computing. The corresponding open-source software is based on this parallel data structure, and it supports multilevel methods on nested meshes and 2D and 3D as well as in space-time. Here, we present generic results on porous media applications including multilevel preconditioning and multilevel Monte Carlo methods for uncertainty quantification

A New Family of Discontinuous Galerkin Schemes for Diffusion Problems

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/143057/1/6.2017-3444.pd

Local Fourier analysis of multigrid for hybridized and embedded discontinuous Galerkin methods

In this paper we present a geometric multigrid method with Jacobi and Vanka relaxation for hybridized and embedded discontinuous Galerkin discretizations of the Laplacian. We present a local Fourier analysis (LFA) of the two-grid error-propagation operator and show that the multigrid method applied to an embedded discontinuous Galerkin (EDG) discretization is almost as efficient as when applied to a continuous Galerkin discretization. We furthermore show that multigrid applied to an EDG discretization outperforms multigrid applied to a hybridized discontinuous Galerkin (HDG) discretization. Numerical examples verify our LFA predictions

A high-order semi-explicit discontinuous Galerkin solver for 3D incompressible flow with application to DNS and LES of turbulent channel flow

We present an efficient discontinuous Galerkin scheme for simulation of the incompressible Navier-Stokes equations including laminar and turbulent flow. We consider a semi-explicit high-order velocity-correction method for time integration as well as nodal equal-order discretizations for velocity and pressure. The non-linear convective term is treated explicitly while a linear system is solved for the pressure Poisson equation and the viscous term. The key feature of our solver is a consistent penalty term reducing the local divergence error in order to overcome recently reported instabilities in spatially under-resolved high-Reynolds-number flows as well as small time steps. This penalty method is similar to the grad-div stabilization widely used in continuous finite elements. We further review and compare our method to several other techniques recently proposed in literature to stabilize the method for such flow configurations. The solver is specifically designed for large-scale computations through matrix-free linear solvers including efficient preconditioning strategies and tensor-product elements, which have allowed us to scale this code up to 34.4 billion degrees of freedom and 147,456 CPU cores. We validate our code and demonstrate optimal convergence rates with laminar flows present in a vortex problem and flow past a cylinder and show applicability of our solver to direct numerical simulation as well as implicit large-eddy simulation of turbulent channel flow at $Re_{\tau}=180$ as well as $590$.Comment: 28 pages, in preparation for submission to Journal of Computational Physic