1549301742-The_Theory_of_Difference_Schemes__Samarskii

116 Basic Concepts of the Theory of Difference Schemes On account of the Cauchy-Bunyakovskil inequality, l(f, y). I= 1(77, Y,,)I ...

Difference sche1nes as operator equations. General formulations 117 Difference schemes as operator equations. After replacing d ...

118 Basic Concepts of the Theory of Difference Schemes The right-hand side vector<[>= (<p 1 , <p 2 , .•• , <pN_ 1 ...

Difference sche1nes as operator equations. General formulations 119 The operator A is positive definite, that is, (Ay, y) > 8 ...

120 Basic Concepts of the Theory of' Difference Sche1nes In this case we write, as before, A = Ah with further reference that h ...

Difference schemes as operator equations. General formulations 121 and (y, w) = Lt,:~^1 Yi wi h. Making use of the Green formula ...

122 Basic Concepts of the Theory of Difference Schemes Likewise, one can prove another inequality By the same token, y^2 (x;) &l ...

Difference sche1nes as operator equations. General for1nulations 123 Exa1nple 3. Non-self-adjoint operators. Let wh = { X; = i h ...

124 Basic Concepts of the Theory of Difference Schemes They are mutually adjoint, that is, (R 1 y, v) = (y, R 2 v) and The usefu ...

Difference schemes as operator equations. General formulations 125 by the difference operators A 1 y = Yxx + b Yx for b > 0 o ...

126 Basic Concepts of the Theory of Difference Schemes operator) in rectangular domains will be studied by means of similar meth ...

Difference schemes as operator equations. General forinulations 127 Stability of the scheme is to be understood as the property ...

128 Basic Concepts of the Theory of Difference Schemes may be true. Here the meaning of weak stability is that condition (25) sh ...

Difference schemes as operator equations. General formulations 129 Remark 1 Convergence conditions (27) imply that (29) lim II Y ...

130 Basic Concepts of the Theory of Difference Schemes Let Yh be a solution of problem (21). We say that 1) scheme (21) converge ...

Difference schemes as operator equations. General formulations 131 of approximation, that is, approximation and col'l'ectness (s ...

132 Basic Concepts of the Theory of Difference Schemes provided conditions (35)-(36) hold. From such reasoning it seems clear th ...

Difference schemes as operator equations. General formulations 133 where A is a linear bounded operator defined in a real Hilber ...

134 Basic Concepts of the Theory of Difference Schemes To make sure of it, we take the inner product of (37) and y = A-^1 <p: ...

Difference schemes as operator equations. General formulations 135 5) Assuming A to be a non-self-adjoint operator subject to th ...