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A well-known and widely applied method of approximating the solutions of problems in mathematical physics is the method of difference schemes. Modern computers allow the implementation of highly accurate ones; hence, their construction and investigation for various boundary value problems in mathematical physics is generating much current interest. The present monograph is devoted to the construction of highly accurate difference schemes for parabolic boundary value problems, based on Padé approximations. The investigation is based on a new notion of positivity of difference operators in Banach spaces, which allows one to deal with difference schemes of arbitrary order of accuracy. Establishing coercivity inequalities allows one to obtain sharp, that is, two-sided estimates of convergence rates. The proofs are based on results in interpolation theory of linear operators. This monograph will be of value to professional mathematicians as well as advanced students interested in the fields of functional analysis and partial differential equations.
List of contents
1 The Abstract Cauchy Problem.- 1. Well-Posedness of the Differential Cauchy Problem in C(E).- 2. Well-Posedness of the Cauchy Problem inC0?(E).- 3. Well-Posedness of the Cauchy Problem in Lp(E).- 4. Well-Posedness of the Cauchy Problem in Lp(E?,Q).- 5. Well-Posedness of the Cauchy Problem in Spaces of Smooth Functions.- 2 The Rothe Difference Scheme.- 0. Stability of the Difference Problem.- 1. Well-Posedness of the Difference Problem in C(E).- 2. Well-Posedness of the Difference Problem in C0?(E).- 3. Well-Posedness of the Difference Problem in Lp(E).- 4. Well-Posedness of the Difference Problem in Lp(E?,Q).- 5. Well-Posedness of the Difference Problem in Difference Analogues of Spaces of Smooth Functions.- 3 PadÉ Difference Schemes.- 0. Stability of the Difference Problem.- 1. Well-Posedness of the Difference Problem in C(E).- 2. Well-Posedness of the Difference Problem in C0?(E).- 3. Well-Posedness of the Difference Problem in Lp(E).- 4. Well-Posedness of the Difference Problem in Lp(E'?,Q).- 5. Well-Posedness of the Difference Problem in Difference Analogues of Spaces of Smooth Functions.- 4 Difference Schemes for Parabolic Equations.- 1. Elliptic Difference Operators with Constant Coefficients.- 2. Fractional Spaces in the case of an Elliptic Difference Operator.- 3. Stability and Coercivity Estimates.- Comments on the Literature.- References.
