Rate of convergence bound of difference schemes for the first boundary-value problem of elasticity theory in arbitrary domains

1993 ◽  
Vol 66 (3) ◽  
pp. 2272-2279
Author(s):  
S. A. Voitsekhovskii ◽  
V. M. Kalinin ◽  
V. L. Makarov
Keyword(s):  
Boundary Value Problem ◽  
Rate Of Convergence ◽  
Elasticity Theory ◽  
Boundary Value ◽  
Difference Schemes ◽  
Convergence Bound

scholarly journals Formal Consistency Versus Actual Convergence Rates of Difference Schemes for Fractional-Derivative Boundary Value Problems

2015 ◽  
Vol 18 (2) ◽  
Author(s):  
José Luis Gracia ◽  
Martin Stynes
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Convergence Rates ◽  
Boundary Value ◽  
Finite Difference Methods ◽  
Polynomial Solution ◽  
Difference Schemes ◽  
Smooth Functions ◽  
Numerical Performance ◽  
The Relationship

AbstractFinite difference methods for approximating fractional derivatives are often analyzed by determining their order of consistency when applied to smooth functions, but the relationship between this measure and their actual numerical performance is unclear. Thus in this paper several wellknown difference schemes are tested numerically on simple Riemann-Liouville and Caputo boundary value problems posed on the interval [0, 1] to determine their orders of convergence (in the discrete maximum norm) in two unexceptional cases: (i) when the solution of the boundary-value problem is a polynomial (ii) when the data of the boundary value problem is smooth. In many cases these tests reveal gaps between a method’s theoretical order of consistency and its actual order of convergence. In particular, numerical results show that the popular shifted Gr¨unwald-Letnikov scheme fails to converge for a Riemann-Liouville example with a polynomial solution p(x), and a rigorous proof is given that this scheme (and some other schemes) cannot yield a convergent solution when p(0)≠ 0.

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