numerical analysis; adaptive mesh refinement; finite element methods; local time-stepping; wave propagation
Grote Marcus J., Nataf Frédéric, Tang Jet Hoe, Tournier Pierre-Henri (2020), Parallel controllability methods for the Helmholtz equation, in Computer Methods in Applied Mechanics and Engineering
, 362, 112846-112846.
Grote Marcus J., Tang Jet Hoe (2019), On controllability methods for the Helmholtz equation, in Journal of Computational and Applied Mathematics
, 358, 306-326.
Grote Marcus J., Mehlin Michaela, Sauter Stefan A. (2018), Convergence Analysis of Energy Conserving Explicit Local Time-Stepping Methods for the Wave Equation, in SIAM Journal on Numerical Analysis
, 56(2), 994-1021.
Wave phenomena are ubiquitous across science, technology and medicine. Typical applications include ultrasound imaging, wireless communications and seismic tomography. In this proposal we shall analyze and further develop time integration methods for the numerical simulation of acoustic, electromagnetic or elastic wave phenomena. For the spatial discretization, we use either conforming finite elements or discontinuous Galerkin methods, which accomodate arbitrary meshes and geometry. For the time discretization, we consider recently derived local time-stepping (LTS) methods, which overcome the bottleneck due to local mesh refinement by taking smaller time-steps precisely where the smallest elements are located.Explicit LTS methods have already proved useful in many applications and shown nearly optimal speed-up on HPC architectures. Convergence (in the ODE sense) to the semi-discrete solution on a fixed mesh is fairly standard. However, a general convergence theory in the PDE sense, which establishes convergence to the (true) continuous solution as both the time-step and the mesh-size tend to zero, is still lacking.This proposal therefore aims at establishing a rigourous convergence theory for explicit LTS methods, which fall into two separate categories. Hence this proposal consists of two separate projects. In the first project, we shall prove optimal space-time convergence of LTS methods based on energy conserving leap-frog (LF) methods. Moreover, we shall compare the accuracy of different fourth-order LTS methods in particular for long-time simulations. In the second project, we shall derive a complete space-time convergence theory for Runge-Kutta (RK) based LTS methods. Moreover, we shall demonstrate the usefulness of LTS methods also in time-harmonic regimes, when the controllability method is used for the solution of the Helmholtz equation.