A posteriori choice of time-discretization step in finite difference methods for solving ill-posed Cauchy problems in Hilbert space

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Mikhail M. Kokurin
Keyword(s):  
Hilbert Space ◽  
Finite Difference ◽  
Finite Difference Methods ◽  
Time Discretization ◽  
Cauchy Problems ◽  
A Posteriori ◽  
Error Level ◽  
Selfadjoint Operators ◽  
Discretization Step ◽  
Ill Posed

Abstract Finite difference semidiscretization methods for solving an ill-posed Cauchy problem in a Hilbert space are investigated. The problems involve linear positively definite selfadjoint operators. We justify an a posteriori scheme for the choice of the time-discretization step and establish accuracy estimates in terms of the error level of input data.

scholarly journals Time discretization of nonlinear Cauchy problems applying to mixed hyperbolic-parabolic equations

1996 ◽  
Vol 19 (3) ◽  
pp. 481-494 ◽  
Author(s):  
Pierluigi Colli ◽  
Angelo Favini
Keyword(s):  
Hilbert Space ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Maximal Monotone ◽  
Time Discretization ◽  
Cauchy Problems ◽  
Initial Value ◽  
Selfadjoint Operators ◽  
Convergence Results ◽  
Uniqueness Of The Solution

In this paper we deal with the equationL(d2u/dt2)+B(du/dt)+Au∋f, whereLandAare linear positive selfadjoint operators in a Hilbert spaceHand from a Hilbert spaceV⊂Hto its dual spaceV′, respectively, andBis a maximal monotone operator fromVtoV′. By assuming some coerciveness onL+BandA, we state the existence and uniqueness of the solution for the corresponding initial value problem. An approximation via finite differences in time is provided and convergence results along with error estimates are presented.

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