Adaptive discontinuous Galerkin methods for elliptic interface problems

Abstract

An interior-penalty discontinuous Galerkin (dG) method for an elliptic interface problem involving, possibly, curved interfaces, with flux-balancing interface conditions, e.g., modelling mass transfer of solutes through semipermeable membranes, is considered. The method allows for extremely general curved element shapes employed to resolve the interface geometry exactly. A residual-type a posteriori error estimator for this dG method is proposed and upper and lower bounds of the error in the respective dG-energy norm are proven. The a posteriori error bounds are subsequently used to prove a basic a priori convergence result. The theory presented is complemented by a series of numerical experiments. The presented approach applies immediately to the case of curved domains with non-essential boundary conditions, too.

A stabilized finite element method for a fictitious domain problem allowing small inclusions

“…We start the error analysis by stating the following local trace inequality. The proof of this result is similar to the one given in [20] for the case of curved elements. Lemma 3.1 LetB i be the neighborhood defined in (3.1).…”

An Unfitted Hybrid High-Order Method with Cell Agglomeration for Elliptic Interface Problems

“…Concerning , we notice that we are using here moderate polynomial degrees (up to 3 in our experiments), and that another important issue to be addressed with the use of high-order methods is a finer control of quadrature errors on the cut cells, as further discussed in Section 5. Interestingly, we point out that under suitable (mild) assumptions on the cut cell geometry, inverse and trace inequalities with optimal dependence on the polynomial degree can be proven; see [13,14].…”

ℎ𝑝-version discontinuous Galerkin methods on essentially arbitrarily-shaped elements

“…Simultaneously, various classes of fitted and unfitted grid methods for interface or transmission problems exploit generalized concepts of mesh elements in an effort to provide accurate representations of internal interfaces. Several unfitted finite element methods have been proposed in recent years: unfitted finite element methods [9], immersed finite element methods [37,36], virtual element methods [25], unfitted penalty methods [10,48,43,60,23], see also [44] for unfitted discretization of the boundary, cutCell/cutFEM [16,14], and unfitted hybrid high-order methods [15], to name just the few closer to the developments we shall be concerned with below. A central idea in the majority of these methods is the weak imposition of interface conditions in conjunction with some form of penalization, see, e.g., [38], an idea going back to [8].…”

“…The question, therefore, of further extending rigorously the applicability of hpversion IP-dG methods to meshes consisting of curved polygonal/polyhedral elements arises naturally. Indeed, such a development is expected to provide multifaceted advantages compared to current approaches, including, but not restricted to, the treatment of curved interfaces as done, e.g., in [48,43,60,15,23]. For instance, allowing for extremely general curved elements enables the exact representation of curved computational domains, e.g., arising directly from Computer Aided Design programs.…”