In this thesis, we present new superconvergence properties of discontinuous Galerkin (DG) methods for two-dimensional hyperbolic problems. We investigate the superconvergence properties of the DG method applied to scalar first-order hyperbolic partial differential equations on triangular meshes. We study the effect of finite element spaces on the superconvergence properties of DG solutions on three types of triangular elements. Superconvergence is described for structured and unstructured meshes. We show that the DG solution is O(hp+1) superconvergent at Legendre points on the outflow edge on triangles having one outflow edge using three p- degree polynomial spaces. For triangles having two outflow edges the finite element error is O(hp+1) superconvergent at the end points of the inflow edge for an augmented space of degree p. Furthermore, we discovered additional mesh-orientation dependent superconvergence points in the interior of triangles. The dependence of these points on orientation is explicitly given. We also established a global superconvergence result on meshes consisting of triangles having one inflow and one outflow edges. Applying a local error analysis, we construct simple, efficient and asymptotically correct a posteriori error estimates for discontinuous finite element solutions of hyperbolic problems on triangular meshes. A posteriori error estimates are needed to guide adaptive enrichment and to provide a measure of solution accuracy for any numerical method. We develop an inexpensive superconvergence-based a posteriori error estimation technique for the DG solutions of conservation laws. We explicitly write the basis functions for the error spaces corresponding to several finite element solution spaces. The leading term of the discretization error on each triangle is estimated by solving a local problem where no boundary conditions are needed. The computed error estimates are shown to converge to the true error under mesh refinement in smooth solution regions. We further present a numerical study of superconvergence properties for the DG method applied to time-dependent convection problems. We also construct asymptotically correct a posteriori error estimates by solving local hyperbolic problems with no boundary conditions on general unstructured meshes. The global superconvergence results are numerically confirmed. Finally, the a posteriori error estimates are tested on several linear and nonlinear problems to show their efficiency and accuracy under mesh refinement. / Ph. D.

We propose a new discontinuous finite element method for higher-order initial value problems where the finite element solution exhibits an optimal convergence rate in the L2- norm. We further show that the q-degree discontinuous solution of a differential equation of order m and its first (m-1)-derivatives are strongly superconvergent at the end of each step. We also establish that the q-degree discontinuous solution is superconvergent at the roots of (q+1-m)-degree Jacobi polynomial on each step. Furthermore, we use these results to construct asymptotically correct a posteriori error estimates. Moreover, we design a new discontinuous Galerkin method to solve the wave equation by using a method of lines approach to separate the space and time where we first apply the classical finite element method using p-degree polynomials in space to obtain a system of second-order ordinary differential equations which is solved by our new discontinuous Galerkin method. We provide an error analysis for this new method to show that, on each space-time cell, the discontinuous Galerkin finite element solution is superconvergent at the tensor product of the shifted roots of the Lobatto polynomials in space and the Jacobi polynomial in time. Then, we show that the global L2 error in space and time is convergent. Furthermore, we are able to construct asymptotically correct a posteriori error estimates for both spatial and temporal components of errors. We validate our theory by presenting several computational results for one, two and three dimensions. / Ph. D.

Purely numerical methods based on the Finite Element Method (FEM) are becoming increasingly popular in seismic modeling for the propagation of acoustic and elastic waves in geophysical models. These methods o er a better control on the accuracy and more geometrical exibility than the Finite Di erence methods that have been traditionally used for the generation of synthetic seismograms. However, the success of these methods has outpaced their analytic validation. The accuracy of the FEMs used for seismic wave propagation is unknown in most cases and therefore the simulation parameters in numerical experiments are determined by empirical rules. I focus on two methods that are particularly suited for seismic modeling: the Spectral Element Method (SEM) and the Interior-Penalty Discontinuous Galerkin Method (IP-DGM). The goals of this research are to investigate the grid dispersion and stability of SEM and IP-DGM, to implement these methods and to apply them to subsurface models to obtain synthetic seismograms. In order to analyze the grid dispersion and stability, I use the von Neumann method (plane wave analysis) to obtain a generalized eigenvalue problem. I show that the eigenvalues are related to the grid dispersion and that, with certain assumptions, the size of the eigenvalue problem can be reduced from the total number of degrees of freedom to one proportional to the number of degrees of freedom inside one element. The grid dispersion results indicate that SEM of degree greater than 4 is isotropic and has a very low dispersion. Similar dispersion properties are observed for the symmetric formulation of IP-DGM of degree greater than 4 using nodal basis functions. The low dispersion of these methods allows for a sampling ratio of 4 nodes per wavelength to be used. On the other hand, the stability analysis shows that, in the elastic case, the size of the time step required in IP-DGM is approximately 6 times smaller than that of SEM. The results from the analysis are con rmed by numerical experiments performed using an implementation of these methods. The methods are tested using two benchmarks: Lamb's problems and the SEG/EAGE salt dome model. / text

Title: Application of hp-adaptive discontinuous Galerkin method to com- pressible flow simulation Author: Karol Tarčák Department: Department of Numerical Mathematics Supervisor: prof. RNDr. Vít Dolejší, Ph.D., DSc. Abstract: In the present work we study an residuum estimate of disconti- nuous Galerkin method for the solution of Navier-Stokes equations. Firstly we summarize the construction of the viscous compressible flow model via Navier-Stokes partial differential equation and discontinuous Galerkin met- hod. Then we propose an extension of an already known residuum estimate for stationary problems to non-stationary problems. We observe the beha- vior of the proposed estimate and modify an existing hp-adaptive algorithm to use our estimate. Finally we apply the modified algorithm on test cases and present adapted meshes from the numerical experiments. Keywords: discontinuous Galerkin method, adaptivity, error estimate 4

Efficient numerical methods for option pricing is an active field of research. This project has the goal to examine possible ways to improve an established method of numerical pricing. The method is based on an adaptive finite difference method in price and uses the backwards differentiation formula of order 2, BDF2, in time. The project will focus on improvements to the time integration through implementation of discontinuous Galerkin methods, dG. Empirical convergence and accuracy results are obtained for equidistant dG-methods up to order 3 and performance is compared to BDF2. The dG-methods do not succeed in outperforming the BDF2-method when comparing accuracy to time for computation, but they do match the performance. Possible ways for improvements are suggested.

We present immersed discontinuous Galerkin finite element methods for one and two dimensional acoustic wave propagation problems in inhomogeneous media where elements are allowed to be cut by the material interface. The proposed methods use the standard discontinuous Galerkin finite element formulation with polynomial approximation on elements that contain one fluid while on interface elements containing more than one fluid they use specially-built piecewise polynomial shape functions that satisfy appropriate interface jump conditions. The finite element spaces on interface elements satisfy physical interface conditions from the acoustic problem in addition to extended conditions derived from the system of partial differential equations. Additional curl-free and consistency conditions are added to generate bilinear and biquadratic piecewise shape functions for two dimensional problems. We established the existence and uniqueness of one dimensional immersed finite element shape functions and existence of two dimensional bilinear immersed finite element shape functions for the velocity. The proposed methods are tested on one dimensional problems and are extended to two dimensional problems where the problem is defined on a domain split by an interface into two different media. Our methods exhibit optimal $O(h^{p+1})$ convergence rates for one and two dimensional problems. However it is observed that one of the proposed methods is not stable for two dimensional interface problems with high contrast media such as water/air. We performed an analysis to prove that our immersed Petrov-Galerkin method is stable for interface problems with high jumps across the interface. Local time-stepping and parallel algorithms are used to speed up computation. Several realistic interface problems such as ether/glycerol, water/methyl-alcohol and water/air with a circular interface are solved to show the stability and robustness of our methods. / Ph. D.

We deal with the numerical solution of the Navier-Stokes equations describing a motion of viscous compressible flows. The governing equations are discretized with the aid of discontinuous Galerkin finite element method which is based on a discontinuous piecewise polynomial approximation. The discretizations leads to a large nonlinear algebraic system. In order to solve this system efficiently, we develop the so-called p-multigrid solution strategy which employ as a projec- tion and a restriction operators the L2 -projection in the spaces of polynomial functions on each element separately. The p-multigrid technique is studied, deve- loped and implemented in the code ADGFEM. The computational performance of the method is presented.

A posteriori error estimation is an inseparable component of any reliable numerical method for solving partial differential equations. The aim of the goal-oriented a posteriori error estimates is to control the computational error directly with respect to some quantity of interest, which makes the method very convenient for many engineering applications. The resulting error estimates may be employed for mesh adaptation which enables to find a numerical approximation of the quantity of interest under some given tolerance in a very efficient manner. In this thesis, the goal-oriented error estimates are derived for discontinuous Galerkin discretizations of the linear scalar model problems, as well as of the Euler equations describing inviscid compressible flows. It focuses on several aspects of the goal-oriented error estimation method, in particular, higher order reconstructions, adjoint consistency of the discretizations, control of the algebraic errors arising from iterative solutions of both algebraic systems, and linking the estimates with the hp-anisotropic mesh adaptation. The computational performance is demonstrated by numerical experiments.

In this thesis, we present a simple and efficient \emph{a posteriori} error estimation procedure for a discontinuous finite element method applied to scalar first-order hyperbolic problems on structured and unstructured tetrahedral meshes. We present a local error analysis to derive a discontinuous Galerkin orthogonality condition for the leading term of the discretization error and find basis functions spanning the error for several finite element spaces. We describe an implicit error estimation procedure for the leading term of the discretization error by solving a local problem on each tetrahedron. Numerical computations show that the implicit \emph{a posteriori} error estimation procedure yields accurate estimates for linear and nonlinear problems with smooth solutions. Furthermore, we show the performance of our error estimates on problems with discontinuous solutions. We investigate pointwise superconvergence properties of the discontinuous Galerkin (DG) method using enriched polynomial spaces. We study the effect of finite element spaces on the superconvergence properties of DG solutions on each class and type of tetrahedral elements. We show that, using enriched polynomial spaces, the discretization error on tetrahedral elements having one inflow face, is O(h^{p+2}) superconvergent on the three edges of the inflow face, while on elements with one inflow and one outflow faces the DG solution is O(h^{p+2}) superconvergent on the outflow face in addition to the three edges of the inflow face. Furthermore, we show that, on tetrahedral elements with two inflow faces, the DG solution is O(h^{p+2}) superconvergent on the edge shared by two of the inflow faces. On elements with two inflow and one outflow faces and on elements with three inflow faces, the DG solution is O(h^{p+2}) superconvergent on two edges of the inflow faces. We also show that using enriched polynomial spaces lead to a simpler{a posterior error estimation procedure. Finally, we extend our error analysis for the discontinuous Galerkin method applied to linear three-dimensional hyperbolic systems of conservation laws with smooth solutions. We perform a local error analysis by expanding the local error as a series and showing that its leading term is O( h^{p+1}). We further simplify the leading term and express it in terms of an optimal set of polynomials which can be used to estimate the error. / Ph. D.

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