Construction of a global solution for the one dimensional singularly-perturbed boundary value problem
This provides an improved convergence rate for approximate solutions of singularly-perturbed problems, which is incremental for researchers in numerical analysis.
The paper constructs a global approximate solution for a one-dimensional singularly-perturbed boundary value problem, achieving ε-uniform convergence of order O(ln²N/N²) on [0,1] after a repair step, confirmed by numerical experiments.
We consider an approximate solution for the one-dimensional semilinear singularly-perturbed boundary value problem, using the previously obtained numerical values of the boundary value problem in the mesh points and the representation of the exact solution using Green's function. We present an $\varepsilon$-uniform convergence of such gained the approximate solutions, in the maximum norm of the order $\mathcal{O}\left(N^{-1}\right)$ on the observed domain. After that, the constructed approximate solution is repaired and we obtain a solution, which also has $\varepsilon$--uniform convergence, but now of order $\mathcal{O}\left(\ln^2N/N^2\right)$ on $[0,1].$ In the end a numerical experiment is presented to confirm previously shown theoretical results.