# Finite Elements Methods for the Cavity Problem

# Finite Elements Methods for the Cavity Problem

This chapter concerns the use of edge finite element methods to approximate the cavity problem with mixed boundary data. It provides two proofs of convergence of the discrete solution. First under restrictive assumptions and using duality theory, which uses the important discrete Helmholtz decomposition that decomposes an edge element field into an exactly curl free discrete field, and a discrete divergence free field. A second more general convergence proof based on the discrete compactness property of edge elements and convergence theory for collectively compact operators is also presented. For small wave numbers, the method can loose stability, so stabilization of the variational problem using appropriate discrete divergence terms is discussed. Finally, the finite element approximation of cavity resonance modes is mentioned again, highlighting discrete compactness.

*Keywords:*
error estimates, discrete Helmholtz decomposition, discrete compactness, discrete eigenvalues

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