The effect of quadrature rules on finite element solutions of Maxwell variational problems

Finite-Element Domain Approximation for Maxwell Variational Problems on Curved Domains

“…Then, theorems 5.8 and 5.9 correspond to discrete analogues to theorems 4.18 and 4.20, respectively. Moreover, our findings allow for a straightforward combination with our earlier results in [4], and so theorems 5.12 and 5.13 correspond to fully discrete versions of theorems 5.8 and 5.9, respectively, by incorporating the effects of numerical integration on error convergence rates.…”

“…The estimates in (3.2) follow from norm equivalence over finitedimensional spaces and (3.1), while (3.3) follows from (3.2) by straightforward computation (cf. Lemma 8 in [4]).…”

“…(2.1) is converted into a second order system for E or H by eliminating the remaining field, requiring pointwise invertibility assumptions on either ε or µ depending on the specific choice. Without loss of generality, we follow [4] and consider the system for the electric field only, assuming the existence of a pointwise inverse of µ. Thus,…”

“…As in [4], we introduce 1 T p a family of quasi-uniform straight meshes of D, written τ p hi , with h i > 0 for all i ∈ N h i → 0 as i grows to infinity, constructed by straight tetrahedrons and indexed by their mesh-sizes, i.e., T p := {τ p hi } i∈N . Throughout, τ p h denotes an arbitrary mesh in T p .…”

“…This analysis is classical in the context of the Laplace equation [15] but has not been studied in the Maxwell case. In [4], we showed the effects of numerical integration on the convergence of the curl-conforming finite element method (FEM) for Maxwell variational problems and found necessary conditions on quadrature rules to ensure error convergence rates both in affine and curved meshes. However, we discarded the error terms associated with domain mesh approximation.…”