Coupling of Discontinuous Galerkin Finite Element and Boundary Element Methods

Coupling of Raviart--Thomas and Hybridizable Discontinuous Galerkin Methods with BEM

“…This formulation involves many more degrees of freedom, especially for high values of k. The LDG-BEM formulations in [12] and [13] cannot be hybridized and require working with two fields in the interior domain (q h , u h ) instead of only one field on the skeleton of the triangulation ( u h ). A recent paper [18] contains numerical experiments for coupling of interior penalty DG methods with BEM. (One of the methods proposed in [18] is related but not equal to methods proposed in [12] and [13].)…”

“…A recent paper [18] contains numerical experiments for coupling of interior penalty DG methods with BEM. (One of the methods proposed in [18] is related but not equal to methods proposed in [12] and [13].) In this case, only the field u h shows up in the interior part of the final system, which, in general will have more degrees of freedom than the hybridized case.…”

Analysis of a non-symmetric coupling of Interior Penalty DG and BEM

“…In a recent article, Of, Rodin, Steinbach and Taus [19] propose three discretization methods that combine Interior Penalty Discontinuous Galerkin methods (IPDG) with Boundary Element Methods (BEM). One of the methods falls into the category of non-symmetric coupling of Finite and Boundary Elements, while the other two belong to the general symmetric coupling philosophy.…”

“…A later paper [12] eliminated the need of the mortar variable and many of the theoretical difficulties by demanding that the discontinuous piecewise polynomial functions that approximate the potential in the LDG method become continuous at the coupling interface. In practice this can be enforced using Lagrange multipliers (see some further explanations in [19]). The paper [12] also showed how to generalize to some methods of the IPDG class.…”

“…The off-diagonal terms in (1.1), involving boundary and interior quantities, are actually non-conforming approximations of H −1/2 (Γ) × H 1/2 (Γ) duality products in the variational formulation of the problem. As already intuited in a remark in [19], the variational technique of [20] (essentially integration by parts after recognizing that the exterior potential has to be transmitted to the interior domain) plays a key role in this analysis. However, integration by parts has now to be applied element-by-element and a whole new array of terms have to be bounded below and hidden by carefully tuned weighted Young inequalities in the essential coercivity estimate (given in Theorem 2.1).…”