Asymptotic expansion techniques for singularly perturbed boundary integral equations

Abstract

For the case of singularly perturbed elliptic transmission problems we demonstrate the use of asymptotic expansion techniques both for establishing regularity results for the solution and for deriving a priori error estimates for boundary element Galerkin discretisation. The dependence of the corresponding bounds on the singular perturbation parameter is studied in detail. This dependence clearly manifests itself in numerical experiments.

A Discrete dual projections

Given a triangulation $\varGamma _{h}$ of $\varGamma$ let $S^{0}_{1}(\widehat{\varGamma }_{h})$ be a space of continuous piecewise linear functions on its barycentric refinement $\widehat{\varGamma }_h$ for which nodal values at the barycentres of cells of $\varGamma _{h}$ provide valid degrees of freedom. The construction of suitable $S^{0}_{1}(\widehat{\varGamma }_{h})$ on a triangulated curve is described in [9, Sect. 4.4.1], and in [3, Sect. 2] for a triangulated surface. In each case simple local computations establish

$$\begin{aligned} \sup \limits _{v_{h}\in S^{0}_{1}(\widehat{\varGamma }_{h})\setminus \{0\}} \frac{|\left<\psi _{h},v_{h}\right>|}{\left\| v_{h}\right\| _{L^2({\varGamma })}} \ge C_{\mathrm {ST}} \left\| \psi _{h}\right\| _{L^2({\varGamma })}\quad \forall \psi _{h}\in S^{-1}_{0}(\varGamma _{h})\;, \end{aligned}$$

(43)

with a constant depending only on the shape-regularity of $\varGamma _{h}$. As an immediate consequence of (43) we have the dual inf-sup condition

$$\begin{aligned} \sup \limits _{\psi _{h}\in S^{-1}_{0}({\varGamma }_{h})\setminus \{0\}} \frac{|\left<\psi _{h},v_{h}\right>|}{\left\| \psi _{h}\right\| _{L^2({\varGamma })}} \ge C_{\mathrm {ST}} \left\| v_{h}\right\| _{L^2({\varGamma })}\quad \forall v_{h}\in S^{0}_{1}(\widehat{\varGamma }_{h})\;. \end{aligned}$$

(44)

Thus we can define two projectors $Q_{h}:L^2({\varGamma })\rightarrow S^{-1}_{0}(\varGamma _{h})$ and $Q_{h}^{*}:L^2({\varGamma })\rightarrow S^{0}_{1}(\widehat{\varGamma }_{h})$ through

$$\begin{aligned} \left<Q_{h}\phi ,v_{h}\right> = \left<\phi ,v_{h}\right>\quad \forall v_{h}\in S^{0}_{1}(\widehat{\varGamma }_{h}),\quad \left<\psi _{h},Q_{h}^{*}v\right> = \left<\psi _{h},v\right>\quad \forall \psi _{h}\in S^{-1}_{0}(\varGamma _{h}).\nonumber \\ \end{aligned}$$

(45)

Both, $Q_{h}$ and $Q_{h}^{*}$ will be $L^2({\varGamma })$-continuous with norms bounded by $C_{\mathrm {ST}}^{-1}$. Moreover,

$$\begin{aligned} \left\| Q_{h}^{*}v\right\| _{H^1({\varGamma })} \le C \left\| v\right\| _{H^1({\varGamma })}\quad \forall v\in H^1({\varGamma })\;, \end{aligned}$$

(46)

where $C>0$ may also depend on the quasi-uniformity of $\varGamma _{h}$. This estimate is a consequence of the continuity of the $L^2({\varGamma })$-orthogonal projection onto $S^{0}_{1}(\widehat{\varGamma }_{h})$ in $H^1({\varGamma })$. Then, interpolation between $H^1({\varGamma })$ and $L^2({\varGamma })$ immediately yields

(47)

Next, we appeal to the definition of the norm of and get

(48)

Thus we have established the following result.

Lemma 5

The projection $Q_{h}$ defined in (45) can be extended to a bounded operator .

We remark that with the same arguments, this result can also be established for the standard $L^2({\varGamma })$-orthogonal projection onto $S^{0}_{1}(\varGamma _{h})$.