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Zbl 1140.65073
Ashyralyev, Allaberen; Cuevas, Claudio; Piskarev, Sergey
On well-posedness of difference schemes for abstract elliptic problems in $L^{p}([0, T];E)$ spaces. (English)
[J] Numer. Funct. Anal. Optim. 29, No. 1-2, 43-65 (2008). ISSN 0163-0563; ISSN 1532-2467/e

Summary: This paper is devoted to the numerical analysis of abstract elliptic differential equations in $L^p([0, T];E)$ spaces. The presentation uses general approximation scheme and is based on $C_{0}$-semigroup theory and a functional analysis approach. For the solutions of difference scheme of the second-order accuracy, the almost coercive inequality in $L^p_{\tau _n}([0, T];E_n)$ spaces with the factor $\min \{\ln \frac {1}{\tau _n}| , 1+ | \ln \parallel B_n \parallel _{B(E_n)}|\}$ is obtained. In the case of UMD space $E_n$, we establish a coercive inequality for the same scheme in $L^p_{\tau _n}([0, T];E_n)$ under the condition of $R$-boundedness.

MSC 2000:: *65N06 Finite difference methods (BVP of PDE)
65J10 Equations with linear operators (numerical methods)
47F05 Partial differential operators
35J25 Second order elliptic equations, boundary value problems
47C10 Operators in *-algebras

Keywords: abstract differential equations; abstract elliptic problem; analytic $C_{0}$-semigroups; Banach spaces; coercive inequality; difference schemes; discrete semigroups; maximal regularity; semidiscretization; UMD spaces; well-posedness

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