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Power series expansion

From the knowledge of the power series expansion of $ u_\varepsilon$ and the computation of several solutions $ u_\varepsilon$ for not too small coefficients $ \varepsilon>0$ , it is possible to approximate the asymptotic solution $ u_0$ [18]:

Theorem 2.4   There exist a constant $ \varepsilon_R>0$ and a family of functions $ (u_n)_{n\in\mathbb{N}}$ of $ H^1(\Omega\backslash\sigma)$ such that for all $ 0\le \varepsilon \le \varepsilon_R$ ,

$\displaystyle u_\varepsilon = \sum_{n=0}^\infty u_n \varepsilon^n.$ (2.48)

Moreover, $ u_0$ is the unique solution in $ H^1(\Omega\backslash\sigma)$ of problem $ (\tilde{\mathcal{P}}_0)$ , and the other functions $ (u_n)$ are the unique solutions in $ H^1(\Omega\backslash\sigma)$ of the following problems:

$\displaystyle (\tilde{\mathcal{P}}_1)\ \ \left\{ \begin{array}{lll} -div\left(\... ... & on & \sigma,\\ \partial_n u_1 = 0 & on & \partial\Omega, \end{array} \right.$ (2.49)

$\displaystyle n\ge 2, (\tilde{\mathcal{P}}_n)\ \ \left\{ \begin{array}{lll} -di... ... & on & \sigma,\\ \partial_n u_n = 0 & on & \partial\Omega. \end{array} \right.$ (2.50)

We can define a function of $ \varepsilon\in\mathbb{R}^+$ as follows

$\displaystyle f(\varepsilon) := u_\varepsilon \in H^1(\Omega\backslash\sigma).$ (2.51)

From the previous theorem, we know that $ f$ has a power series expansion at the origin given by (2.48). We consider a family of $ N$ points $ (\varepsilon_i)$ in $ [\varepsilon_c,\varepsilon_R]$ , where $ \varepsilon_c$ is the smallest value of $ \varepsilon$ for which it is easy to numerically compute $ f(\varepsilon)$ , and $ \varepsilon_R$ is smaller than the convergence radius of the power series. We can then compute an interpolation polynomial $ g_N$ of degree $ N-1$ defined by:

$\displaystyle g_N(\varepsilon) = \sum_{i=1}^N \left( \prod_{j=1, j\ne i}^N \fra... ...arepsilon-\varepsilon_j}{\varepsilon_i-\varepsilon_j}\right) u_{\varepsilon_i},$ (2.52)

where $ N$ is the number of points $ \varepsilon_i$ .

The analycity of $ f$ allows us to estimate the approximation error:

$\displaystyle \Vert u_0 - g_N(0)\Vert _{H^1(\Omega\backslash\sigma)} = \mathcal{O}(\varepsilon_c^N).$ (2.53)

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