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

Abstract

We analyze a non-symmetric coupling of interior penalty discontinuous Galerkin and boundary element methods in two and three dimensions. Main results are discrete coercivity of the method, and thus unique solvability, and quasi-optimal convergence. The proof of coercivity is based on a localized variant of the variational technique from [F.-J. Sayas, The validity of Johnson-Nédeléc's BEM-FEM coupling on polygonal interfaces, SIAM J. Numer. Anal., 47 (5): [3451][3452][3453][3454][3455][3456][3457][3458][3459][3460][3461][3462][3463] 2009]. This localization gives rise to terms which are carefully analyzed in fractional order Sobolev spaces, and by using scaling arguments for rigid transformations. Numerical evidence of the proven convergence properties has been published previously.

A non-symmetric coupling of the finite volume method and the boundary element method

“…Among these extensions there are results on the non-symmetric formulation for the potential equation with variable coefficients [OS13,Ste11], non-linearities [AFF + 13, FFKP15], for elasticity [FFKP15,Ste13], and for boundary value problems [GHS12,OS14]. In addition, similar results have been reported on related coupling formulations [AFF + 13, GHS12] and the DG-BEM coupling [HS15]. We want to mention that the counterpart to the non-symmetric coupling is the so called the symmetric coupling first introduces in [Cos87].…”

The Validity of Johnson--Nédélec's BEM--FEM Coupling on Polygonal Interfaces

“…With the same cut-off and compactness argument that is used for Neumann boundary conditions, it is easy to show that this incoming flood of energy can be controlled by discretization; i.e., once the finite and boundary element spaces are refined enough to take care of the area that separates the interface Γ from the region with different diffusivity, stability is restored. This paper motivated some additional work on nonsymmetric coupling of mixed FEM with BEM [13] and discontinuous Galerkin methods (of the interior penalty family) with BEM [8]. The group of Dirk Praetorius in Vienna has given the corresponding a posteriori error estimates of nonsymmetric BEM-FEM formulations and has worked on the associated adaptive algorithms [16,15].…”

Mortar coupling of $hp$-discontinuous Galerkin and boundary element methods for the Helmholtz equation

“…The first analysis appears to be that of the symmetric coupling of the local DG (LDG) method with BEM in [8,24,25]. Generalizations to nonsymmetric couplings have been proposed in [46] and analyzed in [31]. The closely related coupling of finite volume methods with BEM is analyzed in [18][19][20].…”