Seminar in Numerical Analysis: Ilario Mazzieri (MOX - Politecnico di Milano)
The study and development of spectral element (SE) methods for simulating elastic wave propagation in seismic regions has been subjected to a tremendous growth, occurred in the past ten years. SE methods are based on high-order Lagrangian interpolants sampled at the Gauss-Legendre-Lobatto quadrature points, and combine the flexibility of finite elements with the accuracy of spectral techniques. Since they are based on the weak formulation of the elastodynamics equations, they handle naturally both interface continuity and free boundary conditions, allowing very accurate resolutions of evanescent interface and surface waves. Moreover, SE methods retain a high level parallel structure, thus are well suited for massively parallel computations. The main drawback of SE methods is that they usually require a uniform polynomial order on the whole computational domain, and this can lead to an unreasonably large computational effort, in particular in regions where a fine mesh grid is needed already to describe accurately the domain geometry.
Here, we consider a Discontinuous Galerkin (DGSE) and a Mortar (MSE) spectral element methods coupled with the leap-frog time integration scheme to simulate seismic wave propagations in two and three dimensional heterogeneous media. The main advantage with respect to conforming discretizations, as SE method, is that DGSE and MSE discretizations can accommodate discontinuities, not only in the parameters, but also in the wavefield, while preserving the energy. The domain of interest Ω is assumed to be union of polygonal subdomain Ωi. We allow this subdomain decomposition to be geometrically non-conforming. Inside each subdomain Ωi, a conforming high order finite element space associated to a partition Thi(Ωi) is introduced. We consider different polynomial approximation degrees within different subdomains. To handle non-conforming meshes and non-uniform polynomial degrees across ∂Ωi , a DG or a Mortar discretization is considered.
Applications of the DGSE and MSE methods to simulate realistic seismic wave propagation problems are presented.
Joint work with: P.F. Antonietti, A. Quarteroni and F. Rapetti.
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