SIAM Journal on Numerical Analysis · 1988 · 56 citations · 9 references
Natural SciencesComputational EngineeringMathematicsComputational MechanicsFinite Element AnalysisFinite Element MethodsConvergence AnalysisIntegral EquationsNumerical AnalysisNumerical ConvergenceNumerical IntegrationNumerical Methods For Differential EquationsConvergence Rate AnalysisDiscrete Approximation SchemesNumerical Methods For Integral EquationsPiecewise Polynomial ApproximationPiecewise Polynomial InterpolationQuadrature RulesQuadrature TheoryTime Discretization
In recent papers we have considered the numerical solution of the Hammerstein equation \[y(t) = f(t) + \int_a^b {k(t,s)g(s,y(s))ds,} \quad t \in [a,b],\] by a method that first applies the standard collocation procedure to an equivalent equation for $z(t): = g(t,y(t))$, and then obtains an approximation to y by use of the equation \[ y(t) = f(t) + \int_a^b {k(t,s)z(s)ds,} \quad t \in [a,b].\] We study here a discretized version of the above method. This arises when numerical quadrature is used to approximate the definite integrals occurring in the said method. We consider the use of interpolatory quadrature rules, and we seek the discrete collocation approximation to z in certain piecewise-polynomial function spaces. Our principal result gives the precision required of a quadrature rule to guarantee the best possible (super)convergence rate for the discrete approximation to y. In the case where the kernel k is sufficiently smooth, this rate is of the same order as the rate for an approximation to y obtained via the exact method.
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