A Posteriori Error Estimates and Adaptive Mesh Refinement for the Coupling of the Finite Volume Method and the Boundary Element Method

A nonconforming a posteriori estimator for the coupling of cell-centered finite volume and boundary element methods

“…The last term of the indicator measures the error of the Cauchy data through the Calderón system. The local efficiency of our estimator and thus a lower bound for our energy norm error benefits from the results of [16,18]. More precisely, the results in these papers can be transferred directly to our refinement indicator and we only show an upper bound for the quantities, which include the additional discrete interpolation on the coupling boundary.…”

“…We stress that the computation can be done in linear complexity with respect to the number #T of elements and refer to [6] and [12, p. 18] for more details. The computation of u a and ς a in case of a coupling node a ∈ N Γ is more complicated and derived in [18] for Neumann boundary conditions and in [13,16] for a coupling problem; see Fig. 1.…”

“…The first term on the right-hand side can be estimated with [16,Claim 5] using I m u h as discrete solution and the second by (38). For the rest of the quantities of (22) we may apply the proofs of [18, Claim 1 and Claim 2] and of [16,Claim 1 and Claim 4].…”

“…For the rest of the quantities of (22) we may apply the proofs of [18, Claim 1 and Claim 2] and of [16,Claim 1 and Claim 4].…”

“…We use the Morley interpolation of [18] and extend it in an appropriate way for the coupling problem. We stress that the analysis differs substantially from that in [15,16]. More precisely, the proof for reliability, i.e., to show an upper bound for a certain error, relies on two Helmholtz decompositions of the discontinuous interior Morley error which also affects the exterior part.…”