Condensed Galerkin element of degree m for first-order initial-value problem with O(h2m+2) super-convergent nodal solutions

doi:10.1007/s10483-022-2831-6

Abstract

Abstract: A new type of Galerkin finite element for first-order initial-value problems (IVPs) is proposed. Both the trial and test functions employ the same m-degreed polynomials. The adjoint equation is used to eliminate one degree of freedom (DOF) from the test function, and then the so-called condensed test function and its consequent condensed Galerkin element are constructed. It is mathematically proved and numerically verified that the condensed element produces the super-convergent nodal solutions of O(h2m+2), which is equivalent to the order of accuracy by the conventional element of degree m+1. Some related properties are addressed, and typical numerical examples of both linear and nonlinear IVPs of both a single equation and a system of equations are presented to show the validity and effectiveness of the proposed element.

Key words: Galerkin method, finite element method (FEM), condensed element, superconvergence, adjoint operator, initial-value problem (IVP)

2010 MSC Number:

65L60

Si YUAN, Quan YUAN. Condensed Galerkin element of degree m for first-order initial-value problem with O(h^2m+2) super-convergent nodal solutions. Applied Mathematics and Mechanics (English Edition), 2022, 43(4): 603-614.

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References

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Condensed Galerkin element of degree m for first-order initial-value problem with O(h^2m+2) super-convergent nodal solutions

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