Interior penalty discontinuous galerkin methods for electromagnetic and acoustic wave equations

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.