A priori estimates for certain classes of multidimensional difference initial-boundary-value problems (Q1323979)

Let a family \(E_ h\) of Banach spaces be given where \(h\) is a parameter and let \(A_ h\) denote a bounded linear operator, \(A_ h : E_ h \to E_ h\). The author considers the following difference schemes: \(y(t_{k+1}) = y(t_ k) - \tau \sum^{\nu_ 1}_{j = 1} b_ j Y_ j^{(k)} + \tau \sum^{\nu_ 2}_{j=1} b_ j f_ h(t_ k + c_ j \tau)\), \(y(t_ 0) = R_{\tau h} y_{0h}\), where \(t_ k = k \tau\) \((\tau > 0)\), \(k = 0, 1, \dots, y : [0, \infty) \to E_ h\), \(\nu_ 1,\nu_ 2\) are certain natural numbers, \(b_ j\), \(c_ j\) are complex parameters, \(f_ h : [0, \infty) \to E_ h\) and \(R_{\tau h}\) are bounded linear operators from \(E_ h\) to \(E_ h\). The vectors \(Y^{(k)}_ j\) are found from the following system of equations: \[ Y^{(k)}_ j = \beta_ j y(t_ k) - \tau \sum^{\nu_ 1}_{l=1} a_{jl} Y_ j^{(k)} + \tau \sum^{\nu_ 2} _{l=1} a_{jl} f_ h(t_ k + c_ l \tau),\quad j = 1, 2, \dots, \nu_ 1, \] where \(\beta_ j\), \(a_{jl}\) are certain complex numbers. It is assumed that the discretization generated by the above schemes is \(A\)-stable. Under some assumptions on the operators \(A_ h\) (the so-called \([\psi]\)- pseudopositiveness and others) there exists a unique solution of the problem \(dy/dt + A_ h y = f_ h(t)\), satisfying a priori estimates in norms of the spaces \(E_ h\), if \(R_{\tau h} = I\) or \(R_{\tau h} = \chi\), where \(\chi\) is a fixed rational function being bounded in the half-plane \(\{z : \text{Re} z \geq 0\}\) and satisfying some additional conditions. This result is applied to the operator \(-\sum^ n_{j=1} \partial (a_ j (x) \partial x_ j)/ \partial x_ j + c(x)\) (with boundary conditions of the first kind) in the cube \([0,1]^ n\), where \(a_ j \geq \text{const} > 0\), \(c \geq 0\). Introducing a non-uniform grid the author proves the unique solvability and gives a priori estimates for an associated difference problem in the scale of spaces \(L_{ph}\), \(1 \leq p \leq \infty\). Here \(L_{ph}\) are discrete analogues of spaces \(L_ p\). It turns out that many particular important cases such as the method of Radau, Lobatto, \textit{V. A. Vinokurov} and \textit{N. V. Yuvchenko} [Dokl. Akad. Nauk SSSR 284, No. 2, 272-277 (1985; Zbl 0607.65042)] and Gauss- Legendre may be included in the author's considerations.