Project A2 • Numerical methods for wave problems with nontrivial boundary conditions and nonlocal material laws

Despite the fact that the error analysis of such problems mostly use the same ideas, it is usually carried out for a particular combination of wave equation and numerical method. To avoid repetitive work and ease the derivation of new results, we developed the unified error analysis which applies to many different kinds of wave equations and numerical methods.

Research and results

• Unified error analysis of non-conforming space discretizations for wave-type equations. We propose an abstract framework to prove error bounds in the energy norm for non-conforming space discretizations of linear and semi-linear wave equations, which combines a variational approach with semigroup theory. This systematic analysis provides a priori error bounds in terms of interpolation, data and conformity errors of the method. These bounds can be used to derive convergence rates for a large class of equations and their space discretizations in a simple, systematic and modular way ([HHS19, Hip17, Lei17]). Our unified error analysis is also used in projects A3, A4 and A7, e.g., to analyze the error of a heterogeneous multiscale method for Maxwell’s equation [Fre22, HMS19] or for quasilinear wave equations in [HM22, Mai22]. In addition, the unified error analysis serves as a first step to analyze fully discrete approximations obtained by the method of lines. As examples, we consider the Crank-Nicolson method [Hip17] and algebraically stable and coercive Runge-Kutta methods [Lei21] and show that the full discretization error is bounded by the sum of the (stiff) time integration error and the space discretization error.
• $L^2$ error estimates for space discretizations of wave equations require different techniques than bounds in the energy norm. In [HK21], we show semi-discrete convergence estimates in $L^2$ norm, using non-conforming finite elements for four representative examples of wave equations with dynamic boundary conditions, working in the unified setting as long as the analysis allows. Fully discrete $L^2$ norm convergence results, using algebraically stable and coercive Runge-Kutta methods, are also shown.
• Efficient time integration. We aim at designing time integration schemes that exploit the structure of the corresponding problems. These methods improve the efficiency of standard methods such as, e.g., the implicit Crank-Nicolson method or the explicit leapfrog scheme. For the efficient time integration of problems with locally Lipschitz continuous nonlinearities we proposed a second-order implicit-explicit (IMEX) scheme that treats the linear part of the equation implicitly and the nonlinear part explicitly [HL21]. We proved that the time step size is not restricted by the spatial mesh width but only by the nonlinear part. In order to ensure the stability of the numerical scheme for wave equations with fully nonlinear dynamic boundary conditions, we discretized the nonlinearities in such a way that the quasi-monotonicity of the continuous operators is preserved. Furthermore, we managed to bound the discretization errors of the nonlinearities. While both are straightforward for conforming discretizations, it turns out to be much more involved in the non-conforming case.
• Energy norm error estimates for wave equations with kinetic or acoustic boundary conditions. Since these problems are stated on smooth domains, we consider isoparametric finite element approximations for the space discretization. Using the unified error analysis, we prove convergence rates in an energy norm for the wave equation with acoustic boundary conditions in [HHS19] and kinetic boundary conditions in [Hip17]. Moreover, we provide a rigorous error analysis of the wave equation with semi-linear kinetic boundary conditions [HL20]. We proved novel optimal full discretization error bounds in the energy norm for wave equations with nonlinear damped kinetic boundary conditions in [Lei21] and with acoustic boundary conditions with nonlinear coupling in [Lei22].

Another effective model of interest has been derived in the study of strongly absorbing materials, originally formulated by Leontovich in the 1930s to study the interaction of the earth's surface with radio waves, reads in the time-domain \begin{align*} \partial_{\nu} u &=\dfrac{1}{\varepsilon}\partial_t^{1/2} u \quad\text{ on }\Gamma. \end{align*} Further, we studied acoustic boundary conditions. All of these boundary conditions were considered in the context of wave scattering, namely on an exterior (unbounded) domain, which is the complement of a bounded scatterer. An incoming wave, initially away from the boundary, illuminates a medium at rest and interacts with the material.

To give a complete numerical analysis of problems of this type, the following steps are taken:

• stable time-dependent boundary integral equations (BIEs) are derived and analyzed
• the convolution quadrature method is employed to discretize the BIEs in time
• the boundary element method is employed to discretize the BIEs in space
• an error analysis is conducted, either based on Laplace domain techniques or on energy techniques

Research and results

• In [BLN22], the boundary conditions described above have been treated in the context of acoustic scattering. The boundary integral equations are discretized by polynomial boundary elements and the convolution quadrature method based on multistep methods. Error bounds are proved and illustrated by numerical experiments.
• The electromagnetic counterparts of the described boundary conditions appear in many applications. In [NKL22], we transfer and expand the results of the acoustic setting to the electromagnetic setting. Along the way, several new fundamental time-harmonic bounds are shown. Applying the Runge–Kutta based convolution quadrature method combined with a Raviart–Thomas boundary element discretization yields stable and convergent schemes for the corresponding boundary conditions. Numerical experiments demonstrate the use of the proposed method.
• Finally, in [Nic22] we considered a class of nonlinear boundary conditions. Whereas the boundary integral equation and the corresponding full discretization follows from the previous paper [NKL22], the (nonlinear) error analysis is fundamentally different. Based on energy techniques relying on time-discrete transmission problems error bounds are shown, by leveraging several new innovations in the error analysis. Numerical experiments illustrate the theoretical results and show the effectiveness of the method.

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Particular difficulties of the analysis lay in finding the suitable functional analytic setting which permits the right definition of boundary integral operators, proving the coercivity of the Calderon operator, and ensuring stability of the numerical methods.

Research and results

• Maxwell's equations [KL17]: The suitable abstract framework, allowing a boundary integral formulation, is developed. We show the coercivity of the Calderon operator, upon which the stability of the scheme relies. We prove stability and convergence of a numerical method, using a discontinuous Galerkin method (cf. [HS19] from A4) and leapfrog scheme in the interior coupled to boundary elements and (BDF2) convolution quadrature on the boundary.
• Elastodynamics [Ebe18]: Again the key issue is finding the coercivity of the Calderon operator encoding the transparent boundary conditions, and the abstract framework, which allows to prove its coercitvity. Stability and convergence of a numerical scheme, coupling finite elements and leapfrog time-stepping in the interior with boundary elements and (BDF2) convolution quadratures on the boundary, is proved. Numerical experiments and the handling of implementation issues involving the hyper-singular operator are presented in [Ebe19].
• Runge—Kutta coercivity [BL19]: The coercivity property plays a central role in showing stability of the interor—exteror coupling. Convolution quadratures based on A-stable multistep methods are known to preserve this property. In this paper we study the question: “which Runge—Kutta-based convolution quadratures inherit this property?” It is shown that this holds without any restriction for the third-order Radau IIA method, and on permitting a shift in the Laplace domain variable, this holds for all algebraically stable Runge–Kutta methods and hence for methods of arbitrary order. The results are illustrated with various numerical experiments.

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