ABSTRACT
Positive definite problems enjoy an unconditional discrete stability — the Galerkin method delivers the best approximation in terms of the energy norm. In contrast, Helmholtz-like equations representing wave propagation and vibration problems are only asymptotically stable. Nevertheless, for meshes that are fine enough, the Galerkin method is again (asymptotically) optimal in the energy norm. The situation for mixed problems is quite different. Discrete stability is no longer implied by the continuous stability. Necessary for the discrete stability is the so-called Babusˇka-Brezzi or inf-sup condition relating spaces Q and W and the constraint sesquilinear form d(q , F) involved in mixed formulation (Equation 17.13) (see Exercise 17.5),
sup F∈Q
|d(q , F)| ‖F‖Q ≥ β‖q‖W, ∀q ∈ W,
where constant β is mesh independent, i.e., it depends neither on element size h nor element order p. The condition is trivially satisfied at the continuous level, because the space W is a subspace of space Q (the supremum is attained by F = ∇q ); gradients of potentials from H1 are automatically elements of H(curl). Indeed, the components of the gradient are square integrable by assumption, and the curl of a gradient simply vanishes, so the condition on the square integrability of the curl of a gradient is trivially satisfied.
