# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Well-posedness of the difference schemes for elliptic equations in $C_\tau^{\beta,\gamma}(E)$ spaces. (English) Zbl 1166.65023
The paper deals with a second order of accuracy difference scheme for the periodic (nonlocal) boundary value problem $$-v''(t)+Av(t)=f(t), \; t \in (0,1),$$ $$v(0)=v(1), \; v^{\prime}(0)=v^{\prime}(1)$$ with a strongly positive operator coefficient $A$ in a Banach space. A series of coercivity inequalities in difference analogues of various Hölder norms is obtained and the well-posedness of the difference scheme in $C_{\tau}^{\beta, \gamma}(E)$ spaces is proved. An example of an elliptic $2m$-order multidimensional partial differential equation is considered.

##### MSC:
 65J10 Equations with linear operators (numerical methods) 65N06 Finite difference methods (BVP of PDE) 65N12 Stability and convergence of numerical methods (BVP of PDE) 34G10 Linear ODE in abstract spaces 35J40 Higher order elliptic equations, boundary value problems
Full Text:
##### References:
 [1] Ladyzhenskaya, O. A.; Ural’tseva, N. N.: Linear and quasilinear equations of elliptic type, (1973) · Zbl 0269.35029 [2] Vishik, M. L.; Myshkis, A. D.; Oleinik, O. A.: Partial differential equations, Mathematics in USSR in the last 40 years, 1917--1957 1, 563-599 (1959) [3] Grisvard, P.: Elliptic problems in nonsmooth domains, (1986) · Zbl 0622.34066 [4] Krein, S. G.: Linear differential equations in Banach space, (1966) · Zbl 0168.10801 [5] Gorbachuk, V. L.; Gorbachuk, M. L.: Boundary value problems for differential-operator equations, (1984) · Zbl 0567.47041 [6] Ashyralyev, A.: On well-posedness of the nonlocal boundary value problem for elliptic equations, Numer. funct. Anal. optim. 24, No. 1--2, 1-15 (2003) · Zbl 1055.35018 · doi:10.1081/NFA-120020240 [7] Sobolevskii, P. E.: On elliptic equations in a Banach space, Diff. uravn. 4, No. 7, 1346-1348 (1969) [8] Sobolevskii, P. E.: The coercive solvability of difference equations, Dokl. acad. Nauk SSSR 201, No. 5, 1063-1066 (1971) · Zbl 0246.39002 [9] Ashyralyev, A.; Altay, N.: A note on the well-posedness of the nonlocal boundary value problem for elliptic difference equations, App. math. And comp. 175, No. 1, 49-60 (2006) · Zbl 1094.39004 · doi:10.1016/j.amc.2005.07.013 [10] Smirnitskii, Yu.A.; Sobolevskii, P. E.: Positivity of multidimensional difference operators in the C-norm, Uspekhi mat. Nauk 36, No. 4, 202-203 (1981) [11] Ashyralyev, A.; Sobolevskii, P. E.: New difference schemes for partial differential equations, (2004) · Zbl 1060.65055 [12] Clement, Ph.; Guerre-Delabrire, S.: On the regularity of abstract Cauchy problems and boundary value problems, Math. appl. 9, No. 4, 245-266 (1999) · Zbl 0928.34042 [13] Agarwal, R.; Bohner, M.; Shakhmurov, V. B.: Maximal regular boundary value problems in Banach-valued weighted spaces, Bound. value. Probl. 1, 9-42 (2005) · Zbl 1081.35129 · doi:10.1155/BVP.2005.9 [14] Ashyralyev, A.; Sobolevskii, P. E.: Well-posedness of parabolic difference equations, (1994) · Zbl 1077.39015