Abstract
The authors and their colleagues have developed numerical verification methods for solutions of second-order elliptic boundary value problems based on the infinite-dimensional fixed-point theorem using the Newton-like operator with appropriate approximation and constructive a priori error estimates for Poisson's equations. Many verification results show that the authors' methods are sufficiently useful when the equation has no first-order derivative. However, in the case that the equation includes the term of a first-order derivative, there is a possibility that the verification algorithm does not work even though we adopt a sufficiently accurate approximation subspace. The purpose of this paper is to propose an alternative method to overcome this difficulty. Numerical examples which confirm the effectiveness of the new method are presented.
Similar content being viewed by others
References
R.A. Adams, Sobolev Spaces (Academic Press, New York, 1975).
P.G. Ciarlet, The Finite Element Method for Elliptic Problems (North-Holland, Amsterdam, 1978).
P. Grisvard, Elliptic Problems in Nonsmooth Domains (Pitman, London, 1985).
M.T. Nakao, A numerical approach to the proof of existence of solutions for elliptic problems, Japan J. Appl. Math. 5 (1988) 313–332.
M.T. Nakao, A numerical verification method for the existence of weak solutions for nonlinear boundary value problems, J. Math. Anal. Appl. 164 (1992), 489–507.
M.T. Nakao, Numerical verification methods for solutions of ordinary and partial differential equations, Numer. Funct. Anal. Optim. 22 (2001) 321–356.
M.T. Nakao, N. Yamamoto and S. Kimura, On best constant in the optimal error estimates for the H 10 -projection into piecewise polynomial spaces, J. Approx. Theory 93 (1998) 491–500.
M. Plum, Computer-assisted enclosure methods for elliptic differential equations, Linear Algebra Appl. 324 (2001) 147–187.
S.M. Rump, On the solution of interval linear systems, Computing 47 (1992) 337–353.
S.M. Rump, Verification methods for dense and sparse systems of equations, in: Topics in Validated Computations, ed. J. Herzberger (Elsevier Science/North-Holland, Amsterdam, 1994) pp. 63–135.
Sun Microsystems, Fortran 95 interval arithmetic programming reference, http://docs.sun. com/source/816–2462/index.html.
K. Toyonaga, M.T. Nakao and Y. Watanabe, Verified numerical computations for multiple and nearly multiple eigenvalues of elliptic operators, J. Comput. Appl. Math. 147 (2002) 175–190.
Y. Watanabe, N. Yamamoto, M.T. Nakao and T. Nishida, A numerical verification of nontrivial solutions for the heat convection problem, J. Math. Fluid Mech. 6 (2004) 1–20.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Nakao, M.T., Watanabe, Y. An Efficient Approach to the Numerical Verification for Solutions of Elliptic Differential Equations. Numerical Algorithms 37, 311–323 (2004). https://doi.org/10.1023/B:NUMA.0000049477.75366.94
Issue Date:
DOI: https://doi.org/10.1023/B:NUMA.0000049477.75366.94