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# Generalizations for elliptic reaction-diffusion equations

DOI link for Generalizations for elliptic reaction-diffusion equations

Generalizations for elliptic reaction-diffusion equations book

# Generalizations for elliptic reaction-diffusion equations

DOI link for Generalizations for elliptic reaction-diffusion equations

Generalizations for elliptic reaction-diffusion equations book

## ABSTRACT

Justification of Theorems 2.5.1-2.5.4 on the convergence of the continual and discrete Schwartz method is deduced using their ε-uniform monotonicity. First, we give themonotonicity principle for the continual Schwartz method. Let on the sets D

i ⊂ D (0), for i = 1, . . . , I,with D i = D i(2.91), that form a covering of the set D

(2.90), functions w (k) i (x), x ∈ D

i , where i = 1, . . . , I

and k = 1, 2, 3, . . ., be defined; these functions are 2pi-periodic in xs for s = 2, . . . , n. Let w[0](x), x ∈ D(0), be a prescribed 2pi-periodic function in xs, for s = 2, . . . , n; the functions w[k]i (x), w

[k](x), x ∈ D(0), are defined by the relations

w [k] i (x) = u

( x; {w(k)l (·)}, 1 ≤ l ≤ i, w[k−1](·)

) , i = 1, . . . , I;

w[k](x) = w[k]I (x), x ∈ D (0) , k = 1, 2, 3, . . . .