Well-balanced rr-adaptive and moving mesh space-time discontinuous Galerkin method for the shallow water equations

In this article we introduce a well-balanced discontinuous Galerkin method for the shallow water equations on moving meshes. Particular emphasis will be given on $r$-adaptation in which mesh points of an initially uniform mesh move to concentrate in regions where interesting behaviour of the solution is observed. Obtaining well-balanced numerical schemes for the shallow water equations on fixed meshes is nontrivial and has been a topic of much research. In [S. Rhebergen, O. Bokhove, J.J.W. van der Vegt, Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations, J. Comput. Phys. 227 (2008) 1887–1922] we introduced a well-balanced discontinuous Galerkin method using the theory of weak solutions for nonconservative products introduced in [G. Dal Maso, P.G. LeFloch, F. Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl. 74 (1995) 483–548]. In this article we continue this approach and prove well-balancedness of a discontinuous Galerkin method for the shallow water equations on moving meshes. Numerical simulations are then performed to verify the $r$-adaptive method in combination with the space-time discontinuous Galerkin method against analytical solutions and showing its robustness on more complex problems

Preconditioning of a hybridized discontinuous Galerkin finite element method for the Stokes equations

We present optimal preconditioners for a recently introduced hybridized discontinuous Galerkin finite element discretization of the Stokes equations. Typical of hybridized discontinuous Galerkin methods, the method has degrees-of-freedom that can be eliminated locally (cell-wise), thereby significantly reducing the size of the global problem. Although the linear system becomes more complex to analyze after static condensation of these element degrees-of-freedom, the pressure Schur complement of the original and reduced problem are the same. Using this fact, we prove spectral equivalence of this Schur complement to two simple matrices, which is then used to formulate optimal preconditioners for the statically condensed problem. Numerical simulations in two and three spatial dimensions demonstrate the good performance of the proposed preconditioners

An embedded--hybridized discontinuous Galerkin finite element method for the Stokes equations

We present and analyze a new embedded--hybridized discontinuous Galerkin finite element method for the Stokes problem. The method has the attractive properties of full hybridized methods, namely an $H({\rm div})$-conforming velocity field, pointwise satisfaction of the continuity equation and \emph{a priori} error estimates for the velocity that are independent of the pressure. The embedded--hybridized formulation has advantages over a full hybridized formulation in that it has fewer global degrees-of-freedom for a given mesh and the algebraic structure of the resulting linear system is better suited to fast iterative solvers. The analysis results are supported by a range of numerical examples that demonstrate rates of convergence, and which show computational efficiency gains over a full hybridized formulation

Discontinuous Galerkin finite element methods for (non)conservative partial differential equations

The first research topic in this thesis is the development of discontinuous Galerkin (DG) finite element methods for partial differential equations containing nonconservative products, which are present in many two-phase flow models. For this, we combine the theory of Dal Maso, LeFloch and Murat, in which a definition is given for nonconservative products even where the solution field is discontinuous. This theory also provides the mathematical foundation for a new DG finite element method. For this new DG method, we show standard (p+1)-order convergence results using p-th order basis-functions for test-cases of which we know the exact solution. We also show its ability to deal with more complex test cases. Finally, we apply the method to a depth-averaged two-phase flow model of which the numerical results are qualitatively validated against results obtained from a laboratory experiment. The second topic of this thesis is multigrid. The use of multigrid is of great importance to obtain efficient solvers for fully 3D two-phase flow models. As an initial step to improve the efficiency of solving space-time DG discretizations, we have developed, analyzed and tested optimized multigrid methods using explicit Runge-Kutta type smoothers for the 2D advection-diffusion equation. Many physical models describing fluid motion contain second (and higher) order derivatives. Obtaining a DG discretization for these higher order derivatives is non-trivial and many different DG methods exist to deal with these terms. As final topic of this thesis we introduce an alternative derivation of DG methods based on Borel measures. This alternative derivation gives a consistent treatment of derivative terms by assigning a measure to derivatives when the flow field is discontinuous. We investigate the various DG weak formulations arising from this technique by considering the 2D compressible Navier-Stokes equations for the viscous flows over a cylinder and a NACA0012 airfoil

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

A compatible embedded-hybridized discontinuous Galerkin method for the Stokes--Darcy-transport problem

We present a stability and error analysis of an embedded-hybridized discontinuous Galerkin (EDG-HDG) finite element method for coupled Stokes--Darcy flow and transport. The flow problem, governed by the Stokes--Darcy equations, is discretized by a recently introduced exactly mass conserving EDG-HDG method while an embedded discontinuous Galerkin (EDG) method is used to discretize the transport equation. We show that the coupled flow and transport discretization is compatible and stable. Furthermore, we show existence and uniqueness of the semi-discrete transport problem and develop optimal a priori error estimates. We provide numerical examples illustrating the theoretical results. In particular, we compare the compatible EDG-HDG discretization to a discretization of the coupled Stokes--Darcy and transport problem that is not compatible. We demonstrate that where the incompatible discretization may result in spurious oscillations in the solution to the transport problem, the compatible discretization is free of oscillations. An additional numerical example with realistic parameters is also presented

An interface-tracking space-time hybridizable/embedded discontinuous Galerkin method for nonlinear free-surface flows

We present a compatible space-time hybridizable/embedded discontinuous Galerkin discretization for nonlinear free-surface waves. We pose this problem in a two-fluid (liquid and gas) domain and use a time-dependent level-set function to identify the sharp interface between the two fluids. The incompressible two-fluidd equations are discretized by an exactly mass conserving space-time hybridizable discontinuous Galerkin method while the level-set equation is discretized by a space-time embedded discontinuous Galerkin method. Different from alternative discontinuous Galerkin methods is that the embedded discontinuous Galerkin method results in a continuous approximation of the interface. This, in combination with the space-time framework, results in an interface-tracking method without resorting to smoothing techniques or additional mesh stabilization terms

An exactly mass conserving space-time embedded-hybridized discontinuous Galerkin method for the Navier-Stokes equations on moving domains

This paper presents a space-time embedded-hybridized discontinuous Galerkin (EHDG) method for the Navier--Stokes equations on moving domains. This method uses a different hybridization compared to the space-time hybridized discontinuous Galerkin (HDG) method we presented previously in (Int. J. Numer. Meth. Fluids 89: 519--532, 2019). In the space-time EHDG method the velocity trace unknown is continuous while the pressure trace unknown is discontinuous across facets. In the space-time HDG method, all trace unknowns are discontinuous across facets. Alternatively, we present also a space-time embedded discontinuous Galerkin (EDG) method in which all trace unknowns are continuous across facets. The advantage of continuous trace unknowns is that the formulation has fewer global degrees-of-freedom for a given mesh than when using discontinuous trace unknowns. Nevertheless, the discrete velocity field obtained by the space-time EHDG and EDG methods, like the space-time HDG method, is exactly divergence-free, even on moving domains. However, only the space-time EHDG and HDG methods result in divergence-conforming velocity fields. An immediate consequence of this is that the space-time EHDG and HDG discretizations of the conservative form of the Navier--Stokes equations are energy stable. The space-time EDG method, on the other hand, requires a skew-symmetric formulation of the momentum advection term to be energy-stable. Numerical examples will demonstrate the differences in solution obtained by the space-time EHDG, EDG, and HDG methods

A locally conservative and energy-stable finite element for the Navier--Stokes problem on time-dependent domains

We present a finite element method for the incompressible Navier--Stokes problem that is locally conservative, energy-stable and pressure-robust on time-dependent domains. To achieve this, the space--time formulation of the Navier--Stokes problem is considered. The space--time domain is partitioned into space--time slabs which in turn are partitioned into space--time simplices. A combined discontinuous Galerkin method across space--time slabs, and space--time hybridized discontinuous Galerkin method within a space--time slab, results in an approximate velocity field that is $H({\rm div})$-conforming and exactly divergence-free, even on time-dependent domains. Numerical examples demonstrate the convergence properties and performance of the method

A pressure-robust embedded discontinuous Galerkin method for the Stokes problem by reconstruction operators

The embedded discontinuous Galerkin (EDG) finite element method for the Stokes problem results in a point-wise divergence-free approximate velocity on cells. However, the approximate velocity is not H(div)-conforming and it can be shown that this is the reason that the EDG method is not pressure-robust, i.e., the error in the velocity depends on the continuous pressure. In this paper we present a local reconstruction operator that maps discretely divergence-free test functions to exactly divergence-free test functions. This local reconstruction operator restores pressure-robustness by only changing the right hand side of the discretization, similar to the reconstruction operator recently introduced for the Taylor--Hood and mini elements by Lederer et al. (SIAM J. Numer. Anal., 55 (2017), pp. 1291--1314). We present an a priori error analysis of the discretization showing optimal convergence rates and pressure-robustness of the velocity error. These results are verified by numerical examples. The motivation for this research is that the resulting EDG method combines the versatility of discontinuous Galerkin methods with the computational efficiency of continuous Galerkin methods and accuracy of pressure-robust finite element methods