Interior penalty discontinuous galerkin methods for electromagnetic and acoustic wave equations

Abstract

Introduction: In this thesis we present and analyze the numerical approximation of the second order electromagnetic and acoustic wave equation by the interior penalty (IP) discontinuous Galerkin (DG) finite element method (FEM). In Part I we focus on time-harmonic Maxwell source problems in the high-frequency regime. Part II is devoted to the study of the IP DG FEM for time-dependent acoustic and electromagnetic wave equations. We begin by stating Maxwell's equations in time and frequency domain. We proceed by a variational formulation of Maxwell's equations, and describe the key challenges that are faced in the analysis of the Maxwell operator. Then, we review conforming finite element methods to discretize the second order Maxwell operator. We end this general introduction with some numerical results to highlight the performance and feasibility of conforming FEM for Maxwell's equations. Chapter 2: In this chapter, we introduce and analyze the interior penalty discontinuous Galerkin method for the numerical discretization of the indefinite time-harmonic Maxwell equations in high-frequency regime. Based on suitable duality arguments, we derive a-priori error bounds in the energy norm and the L2-norm. In particular, the error in the energy norm is shown to converge with the optimal order O(hminfs;`g) with respect to the mesh size h, the polynomial degree `, and the regularity exponent s of the analytical solution. Under additional regularity assumptions, the L2-error is shown to converge with the optimal order O(h`+1). The theoretical results are confirmed in a series of numerical experiments on triangular meshes. The thesis' author's principal contributions are the proof of the L2-error bound in Section 2.6, and the proof of Lemma 2.4.1. Chapter 3: We present and analyze an interior penalty method for the numerical discretization of the indefinite time-harmonic Maxwell equations in mixed form. The method is based on the mixed discretization of the curl-curl operator developed in [44] and can be understood as a non-stabilized variant of the approach proposed in [63]. We show the well-posedness of this approach and derive optimal a-priori error estimates in the energy-norm as well as the L2-norm. The theoretical results are confirmed in a series of numerical experiments. The thesis' author's principal contribution is the proof of the L2-error bound in Section 3.6. Chapter 4: The symmetric interior penalty discontinuous Galerkin finite element method is presented for the numerical discretization of the second-order scalar wave equation. The resulting stiffness matrix is symmetric positive definite and the mass matrix is essentially diagonal; hence, the method is inherently parallel and, leads to fully explicit time integration when coupled with an explicit timestepping scheme. Optimal a priori error bounds are derived in the energy norm and the L2-norm for the semi-discrete formulation. In particular, the error in the energy norm is shown to converge with the optimal order O(hminfs;`g) with respect to the mesh size h, the polynomial degree `, and the regularity exponent s of the continuous solution. Under additional regularity assumptions, the L2-error is shown to converge with the optimal order O(h`+1). Numerical results confirm the expected convergence rates and illustrate the versatility of the method. Chapter 5: We develop the symmetric interior penalty discontinuous Galerkin (DG) method for the spatial discretization in the method of lines approach of the timedependent Maxwell equations in second-order form. We derive optimal a-priori estimates for the semi-discrete error in the energy norm. For smooth solutions, these estimates hold for DG discretizations on general finite element meshes. For low-regularity solutions that have singularities in space, the theoretical estimates hold on conforming, affine meshes. Moreover, on conforming triangular meshes, we derive optimal error estimates in the L2-norm. Finally, we valuate our theoretical results by a series of numerical experiments.