Asymptotic and numerical analysis of elliptic-parabolic equation

Authors:

This article is devoted to the boundary value problem for elliptic-parabolic equation with small parameters by the second-order derivatives. The aim of our work is to construct an effective numerical algorithm based on asymptotic approximation for the solution.
We consider the rectangle $\Omega=(0,1)\times(-1,1)$ in $(x,y)$ plain, and denote its upper part as $\Omega_p=(0,1)\times(0,1)$, and its part as $\Omega_e=(0,1)\times(-1,0)$. We introduce the interface $\gamma=(0,1)\times \{0\}$ between these two domains, as well as boundary parts: $\Gamma_p=\{0\}\times(0,1)\cup\{1\}\times(0,1)$;$\Gamma_e=\{0\}\times(-1,0)\cup\{1\}\times(-1,0)$; $\Gamma_D=(0,1)\times\{1\}$.
We consider the mixed type equation which has parabolic type in $\Omega_p$ and elliptic in $\Omega_e$. We pose Dirichlet boundary condition on $\Gamma=\Gamma_e\cup\Gamma_p\cup\Gamma_D$ and transmission condition on $\gamma$:
\begin{equation*}
\begin{cases}
Mu\equiv a_pu+b_pu_y-\varepsilon u_{xx}=f_p, &\text{in $\Omega_p$,}\\
Lu \equiv a_eu+b_eu_y-\varepsilon(u_{xx}+u_{yy})=f_e &\text{in $\Omega_e$,}\\
u=0 &\text{in $\Gamma$,}\\
\text{$u$ and $u_y$ are continuous} &\text{throught $\gamma$.}
\end{cases}
\end{equation*}
Here $\varepsilon$ is small positive parameter, and $a_p, b_p, f_p, a_e, b_e, f_e$ and smooth functions depending on $(x,y)$.
Our work consists of two parts: asymptotic analysis of the problem and creation of effective numerical algorithm based on asymptotic approximation.
In the first part, we apply modification of boundary functions method for mixed type equations, in order to obtain asymptotic representation for our solution with respect to the small parameter.
The main idea of the second part is that numerical calculation of parabolic equation is much easier and requires less operations then the elliptic one. Using again the fact that parameter $\varepsilon$ is small, we construct approximate factorization of elliptic operator, replacing it by the product of two parabolic operators. Instead of one elliptic problem, we calculate numerically two successive parabolic problems: the first problem for inverse parabolic operator in decreasing direction, the second problem for direct parabolic operator in increasing direction. To begin numeric algorithm, we need to calculate initial condition for the first parabolic equation. As it cannot be obtained explicitly, we use its asymptotic representation from the first part. Factorization idea allows us to gain $O(n^2)$ computer operation compared to $O(n^3)$, where $n$ is a number of points of
discretization.
The main advantage of this numerical algorithm is that the problem can be solved faster and demands less computer resources then classical numerical scheme.
Key words: mixed type equation, boundary value, problem, singular perturbations, method of small parameter, operator factorization, numerical algorithm.

UDC:
517.95