Skip to main content
SpringerLink
Log in

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

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. HENRICI, P. Discrete Variable Methods in Ordinary Differential Equations, Wiley, New York (1962)

    MATH  Google Scholar 

  2. CESCHINO, F. and KUNTZMANN, J. Numerical Solution of Initial Value Problems, Prentice-Hall, Englewood Cliffs, New Jersey (1966)

    MATH  Google Scholar 

  3. GEAR, C. W. Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, New Jersey (1971)

    MATH  Google Scholar 

  4. STRANG, G. and FIX, G. J. An Analysis of the Finite Element Method, Prentice-Hall, New York (1973)

    MATH  Google Scholar 

  5. KELLER, H. B. Numerical Solution of Two Point Boundary Value Problems, Society for Industrial and Applied Mathematics, Philadelphia (1976)

    Book  Google Scholar 

  6. FLETCHER, C. Computational Galerkin Methods, Springer-Verlag, New York (1984)

    Book  Google Scholar 

  7. WILLIAM, J. L. Galerkin methods for two-point boundary value problems for first order systems. SIAM Journal on Numerical Analysis, 20, 161–171 (1983)

    Article  MathSciNet  Google Scholar 

  8. DELFOUR, M., HAGER, W., and TROCHU, F. Discontinuous Galerkin methods for ordinary differential equations. Mathematics of Computation, 36, 455–473 (1981)

    Article  MathSciNet  Google Scholar 

  9. HULME, B. L. One-step piecewise polynomial Galerkin methods for initial value problem. Mathematics of Computation, 26, 415–426 (1972)

    Article  MathSciNet  Google Scholar 

  10. DOUGLAS, J. and DUPONT, T. Galerkin approximations for the two point boundary problems using continuous, piecewise polynomial spaces. Numerical Mathematics, 22, 99–109 (1974)

    Article  MathSciNet  Google Scholar 

  11. DOUGLAS, J. and DUPONT, T. Galerkin methods for parabolic equations. SIAM Journal on Numerical Analysis, 7, 575–626 (1970)

    Article  MathSciNet  Google Scholar 

  12. XIAO, C. Adaptive FEM Analysis for Systems of First Order ODEs Based on EEP Superconvergent Method (in Chinese), Ph.D. dissertation, Tsinghua University, Beijing (2009)

    Google Scholar 

  13. YUAN, S., XING, Q. Y., WANG, X., and YE, K. S. Self-adaptive strategy for one-dimensional finite element method based on EEP method with optimal super-convergence order. Applied Mathematics and Mechanics (English Edition), 29(5), 591–602 (2008) https://doi.org/10.1007/s10483-008-0504-8

    Article  MathSciNet  Google Scholar 

  14. YUAN, S., DU, Y., XING, Q. Y., and YE, K. S. Self-adaptive one-dimensional nonlinear finite element method based on element energy projection method. Applied Mathematics and Mechanics (English Edition), 35(10), 1223–1232 (2014) https://doi.org/10.1007/s10483-014-1869-9

    Article  MathSciNet  Google Scholar 

Download references

Funding

Project supported by the National Natural Science Foundation of China (Nos. 51878383 and 51378293)

Author information

Authors and Affiliations

  1. Department of Civil Engineering, Key Laboratory of Civil Engineering Safety and Durability of China Education Ministry, Tsinghua University, Beijing, 100084, China

    Si Yuan & Quan Yuan

Authors
  1. Si Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Quan Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Quan Yuan.

Additional information

Citation: YUAN, S. and YUAN, Q. Condensed Galerkin element of degree m for first-order initial-value problem with O(h2m+2) super-convergent nodal solutions. Applied Mathematics and Mechanics (English Edition), 43(4), 603–614 (2022) https://doi.org/10.1007/s10483-022-2831-6

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yuan, S., Yuan, Q. Condensed Galerkin element of degree m for first-order initial-value problem with O(h2m+2) super-convergent nodal solutions. Appl. Math. Mech.-Engl. Ed. 43, 603–614 (2022). https://doi.org/10.1007/s10483-022-2831-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-022-2831-6

Key words

Chinese Library Classification

2010 Mathematics Subject Classification

Search

Navigation