September 18 – Théophile Chaumont: High order finite element methods for the Helmholtz equation in highly heterogeneous media

Théophile Chaumont: Monday 18 September at 1 pm, A415 Inria Paris.

Time-harmonic wave propagation problems are costly to solve numerically
since the corresponding PDE operators are not strongly elliptic, and as a result,
discretization methods might become unstable. Specifically, the finite element solution is
quasi-optimal (almost as good as the best approximation the finite element space
can provide) only under restrictive assumptions on the mesh size. If the mesh
size is too large, stability is lost, and the finite element solution can become
completely inaccurate, even when the best approximation is. This phenomenon is called the
« pollution effect » and becomes more important for larger frequencies.

For the case of wave propagation problems in homogeneous media, it is known that
high order finite element methods are less sensitive to the pollution effect. For this
reason, they are employed in a wide range of applications, as the corresponding linear
systems are smaller and easier to solve.

In this talk, we investigate the use of high order finite element methods to solve
wave propagation problems in highly heterogeneous media. Since the heterogeneities
of the medium can exhibit small scale features, we consider « non-fitting » meshes,
that are not aligned with the physical interfaces of the medium. Instead, the parameters
defining the medium of propagation can be discontinuous inside each element. We propose
a convergence analysis and draw two main conclusions:
– the asymptotic convergence rate of the proposed finite element method is suboptimal
due to the lack of regularity of the solution inside each cell
– the pollution effect is greatly reduced by increasing the order of discretization.
We illustrate our main conclusions with geophysical application benchmarks.
These examples confirm that higher order methods are more efficient than
linear finite elements.

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