Abstract
A high-order cut finite element method is formulated for solving the elastic wave equation. Both a single domain problem and an interface problem are treated. The boundary or interface is allowed to cut through the background mesh. To avoid problems with small cuts, stabilizing terms are added to the bilinear forms corresponding to the mass and stiffness matrix. The stabilizing terms penalize jumps in normal derivatives over the faces of the elements cut by the boundary/interface. This ensures a stable discretization independently of how the boundary/interface cuts the mesh. Nitsche’s method is used to enforce boundary and interface conditions, resulting in symmetric bilinear forms. As a result of the symmetry, an energy estimate can be made and optimal order a priori error estimates are derived for the single domain problem. Finally, numerical experiments in two dimensions are presented that verify the order of accuracy and stability with respect to small cuts.
Highlights
The time-dependent elastic wave equation is important in several applications
The dynamics of surface and interface waves are directly influenced by for example the curvature, while bulk waves are influenced by the reflection, transmission, and conversion to other types of waves, that occur at surfaces and material interfaces
The mathematical problems are stated. These are the elastic wave equation posed on a single domain and as an interface problem
Summary
The time-dependent elastic wave equation is important in several applications. For example, materials in the Earth’s crust can be modeled as linearly elastic, and earthquakes give rise to seismic waves that propagate over large distances. Immersed methods can avoid complicated mesh generation, and it follows that this approach could potentially be computationally cheaper This argument is strengthened when the geometry of a boundary or interface changes. Since the geometry changes between iterations, a boundary conforming mesh could become very deformed, which would lead to a need for a potentially time-consuming re-meshing procedure This type of inversion problem has been of interest for some time, see for example [29,30,31]. As far as we know, none of these methods have been applied to the time-dependent elastic wave equation We will use another approach to achieve high accuracy, which is to develop a highorder method of Cut-FEM type. The contribution of this paper is an up to fourth-order accurate immersed method of Cut-FEM type for solving the time-dependent elastic wave equation. We define the physical domain and interfaces implicitly by level-set functions, and the needed high-order
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