Finite difference domain decomposition algorithms for a parabolic problem with boundary layers

Parallel algorithms for a singularly perturbed parabolic problem

“…For problem (1), we consider a uniform difference scheme on a nonequidistant mesh from [5,9]. On the setΩ × [0, T ], we introduce a rectangular meshΩ h ×Ω τ , whereΩ h =Ω hx ×Ω hy :…”

“…. Now we introduce a special nonuniform mesh that is adapted to the singularly perturbed behavior of the exact solution [5].…”

“…On each time-step, this approach reduces a given parabolic problem to a sequence of elliptic problems on appropriate subdomains. In [5], finite domain decomposition methods for one-dimensional singularly perturbed parabolic problems have been investigated. Here decomposition of the original computational domain is similar to the approach proposed in [6], in which the domain is partitioned into many nonoverlapping subdomains with interface Γ.…”

“…The rest of this article is organized as follows. In Section II, we consider the undecomposed algorithm from [5,9], which exhibits uniformity in the small parameter convergence. In Section III, we construct two finite domain decomposition algorithms with different time discretizations.…”

A variant of Tikhonov regularization for parabolic PDE with space derivative multiplied by a small parameter∊

“…It is well known that the solution of (1.1) and (1.2) exhibit a boundary layer at x ¼ 0 and x ¼ 1 [15][16][17]. The singularly perturbed time-dependent problems have been an interesting subject for many researchers and many numerical methods were proposed in literature to solve parabolic problems [1][2][3][6][7][8][12][13][14]. Although various numerical schemes are available in literature to find the approximate solution of such problems, the discretization of the PDE often leads to a highly ill-conditioned system and that results in an unstable solution.…”