Partial expansion of a Lipschitz domain and some applications

We show that a Lipschitz domain can be expanded solely near a part of its boundary, assuming that the part is enclosed by a piecewise C1 curve. The expanded domain as well as the extended part are both Lipschitz. We apply this result to prove a regular decomposition of standard vector Sobolev spaces with vanishing traces only on part of the boundary. Another application in the construction of low-regularity projectors into finite element spaces with partial boundary conditions is also indicated

Stabilized mixed approximation of axisymmetric Brinkman flows

This paper is devoted to the numerical analysis of an augmented finite element approximation of the axisymmetric Brinkman equations. Stabilization of the variational formulation is achieved by adding suitable Galerkin least-squares terms, allowing us to transform the original problem into a formulation better suited for performing its stability analysis. The sought quantities (here velocity, vorticity, and pressure) are approximated by Raviart−Thomas elements of arbitrary order k ≥ 0, piecewise continuous polynomials of degree k + 1, and piecewise polynomials of degree k, respectively. The well-posedness of the resulting continuous and discrete variational problems is rigorously derived by virtue of the classical Babuška–Brezzi theory. We further establish a priori error estimates in the natural norms, and we provide a few numerical tests illustrating the behavior of the proposed augmented scheme and confirming our theoretical findings regarding optimal convergence of the approximate solutions

Wavenumber Explicit Analysis of a DPG Method for the Multidimensional Helmholtz Equation

We study the properties of a novel discontinuous Petrov Galerkin (DPG) method for acoustic wave propagation. The method yields Hermitian positive definite matrices and has good pre-asymptotic stability properties. Numerically, we find that the method exhibits negligible phase errors (otherwise known as pollution errors) even in the lowest order case. Theoretically, we are able to prove error estimates that explicitly show the dependencies with respect to the wavenumber ω, the mesh size h, and the polynomial degree p. But the current state of the theory does not fully explain the remarkably good numerical phase errors. Theoretically, comparisons are made with several other recent works that gave wave number explicit estimates. Numerically, comparisons are made with the standard finite element method and its recent modification for wave propagation with clever quadratures. The new DPG method is designed following the previously established principles of optimal test functions. In addition to the nonstandard test functions, in this work, we also use a nonstandard wave number dependent norm on both the test and trial space to obtain our error estimates

Commuting Smoothed Projectors in Weighted Norms with an Application to Axisymmetric Maxwell Equations

We construct finite element projectors that can be applied to functions with low regularity. These projectors are continuous in a weighted norm arising naturally when modeling devices with axial symmetry. They have important commuting diagram properties needed for finite element analysis. As an application, we use the projectors to prove quasioptimal convergence for the edge finite element approximation of the axisymmetric time-harmonic Maxwell equations on nonsmooth domains. Supplementary numerical investigations on convergence deterioration at high wavenumbers and near Maxwell eigenvalues and are also reported