Grid approximation of singularly perturbed boundary value problem for quasi-linear parabolic equations in the case of complete degeneracy in spatial variables

1991 ◽  
Vol 6 (3) ◽  
Author(s):  
G. I. SHISHKIN
Keyword(s):  
Boundary Value Problem ◽  
Parabolic Equations ◽  
Boundary Value ◽  
Singularly Perturbed ◽  
Spatial Variables ◽  
Grid Approximation ◽  
Linear Parabolic Equations

On the solutions of the first boundary value problem for the linear parabolic equations

1988 ◽  
Vol 108 (3-4) ◽  
pp. 339-355 ◽  
Author(s):  
Eugenio Sinestrari ◽  
Wolf von Wahl
Keyword(s):  
Parabolic Equation ◽  
Boundary Value Problem ◽  
Parabolic Equations ◽  
Boundary Value ◽  
Second Order ◽  
Inhomogeneous Term ◽  
Hölder Continuous ◽  
Space And Time ◽  
Order Parabolic Equation ◽  
Linear Parabolic Equations

SynopsisThe first boundary value problem for a linear second order parabolic equation is studied under the assumption that the inhomogeneous term is continuous in space and time and Hölder-continuous only with respect to the space variables.

GRID APPROXIMATION OF SINGULARLY PERTURBED PARABOLIC EQUATIONS WITH MOVING BOUNDARY LAYERS

2008 ◽  
Vol 13 (3) ◽  
pp. 421-442
Author(s):  
Grigorii Shishkin
Keyword(s):  
Boundary Value Problem ◽  
Moving Boundary ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Singularly Perturbed ◽  
Reaction Diffusion Equation ◽  
Positive Direction ◽  
Grid Approximation ◽  
Necessary And Sufficient

A grid approximation of a boundary value problem is considered for a singularly perturbed parabolic reaction‐diffusion equation in a domain with boundaries moving along the x‐axis in the positive direction. For small values of the parameter ϵ (that is the coefficient of the highest‐order derivative in the equation, ϵ ∈ (0,1]), a moving boundary layer appears in a neighbourhood of the left lateral boundary SL 1. It turns out that, in the class of difference schemes on rectangular grids condensing in a neighbourhood of SL 1 with respect to x and t, there do not exist schemes that converge even under the condition P 0 −1 Â ϵ1/2, where P 0 is the total number of nodes in the meshes used, that is, P 0 Â N N 0, where the values N and N 0 define the numbers of mesh points in x and t. On such meshes, convergence under the condition N −1 + N 0 −1 ≤ ϵ1/4 cannot be achieved. Examination of widths similar to Kolmogorov's widths allows us to establish necessary and sufficient conditions for the ϵ‐uniform convergence of approximations to the solution of the boundary value problem. Using these conditions, a scheme is constructed on a mesh being piece‐wise uniform in a coordinate system adapted to the moving boundary. This scheme converges ϵ‐uniformly at the rate O(N −1 ln N + N0 −1).

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