Coupling of the Finite Volume Element Method and the Boundary Element Method: An a Priori Convergence Result

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

“…The given data satisfy f ∈ L 2 (Ω), u 0 ∈ H 1/2 (Γ ) and t 0 ∈ L 2 (Γ ) on Γ . The existence and uniqueness for this model problem is shown, e.g., in [14].…”

“…For a detailed discussion on ξ ∈ H 1/2 * (Γ ), on the choice of a ∞ or b ∞ , and the calculation of the weak form we refer to [13,14] and the references therein.…”

“…A coupling method, which preserves the symmetry of a system, is introduced in [7]. For an FVM-BEM coupling procedure existence, uniqueness and a priori error estimates are given in [14]. There, a vertex-centered FVM approximates a convection diffusion reaction interior problem and is coupled with the BEM, which solves a diffusion process in an unbounded exterior domain.…”

“…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.…”

“…A tangential derivative jump on the coupling boundary measures the error of the tangential derivative between the trace solution of the exterior problem and a piecewise affine and globally continuous solution on the coupling boundary, which is generated from the interior solution. We remark, that the coupling of the cell-centered FVM and the BEM [13,15] has an additional block in the discrete system compared to the vertex centered FVM-BEM [13,14] or a finite element boundary element coupling [4]. This results from further continuous ansatz functions on the interface to link the discontinuous displacement field from the cell-centered FVM to necessarily continuous boundary ansatz functions from the BEM on the coupling boundary.…”