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Mathematics of Computation

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One-step piecewise polynomial Galerkin methods for initial value problems


Author: Bernie L. Hulme
Journal: Math. Comp. 26 (1972), 415-426
MSC: Primary 65L05
DOI: https://doi.org/10.1090/S0025-5718-1972-0321301-2
MathSciNet review: 0321301
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Abstract: A new approach to the numerical solution of systems of first-order ordinary differential equations is given by finding local Galerkin approximations on each subinterval of a given mesh of size $ h$. One step at a time, a piecewise polynomial, of degree $ n$ and class $ {C^0}$, is constructed, which yields an approximation of order $ O({h^{2n}})$ at the mesh points and $ O({h^{n + 1}})$ between mesh points. In addition, the $ j$th derivatives of the approximation on each subinterval have errors of order $ O({h^{n - j + 1}}),1 \leqq j \leqq n$. The methods are related to collocation schemes and to implicit Runge-Kutta schemes based on Gauss-Legendre quadrature, from which it follows that the Galerkin methods are $ A$-stable.


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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1972-0321301-2
Keywords: Galerkin method, initial value problems, ordinary differential equations, piecewise polynomials, Gauss-Legendre quadrature, collocation methods, implicit Runge-Kutta methods, $ A$-stable
Article copyright: © Copyright 1972 American Mathematical Society

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