A posteriori estimates of inverse operators for boundary value problems in linear elliptic partial differential equations

@article{Watanabe2013APE,
  title={A posteriori estimates of inverse operators for boundary value problems in linear elliptic partial differential equations},
  author={Yoshitaka Watanabe and Takehiko Kinoshita and Mitsuhiro T. Nakao},
  journal={Math. Comput.},
  year={2013},
  volume={82},
  pages={1543-1557}
}
This paper presents constructive a posteriori estimates of inverse operators for boundary value problems in linear elliptic partial differential equations (PDEs) on a bounded domain. This type of estimate plays an important role in the numerical verification of the solutions for boundary value problems in nonlinear elliptic PDEs. In general, it is not easy to obtain the a priori estimates of the operator norm for inverse elliptic operators. Even if we can obtain these estimates, they are often… 

Tables from this paper

Some Remarks on the Rigorous Estimation of Inverse Linear Elliptic Operators
TLDR
A new numerical method is presented to obtain the rigorous upper bounds of inverse linear elliptic elliptic operators and it is shown the proposed new estimate is effective for an intermediate mesh size.
An improvement of the theorem of a posteriori estimates for inverse elliptic operators
: This paper presents a numerical method to verify the invertibility of a linear elliptic operator. The invertibility of a linearized operator is useful information when verifying the existence of a
On verified computations of solutions for nonlinear parabolic problems
: We consider the methods for guaranteed computations of solutions for nonlinear parabolic initial-boundary value problems. First, in order to make the basic principle clear, we briefly introduce the
Some remarks on the instability of approximate solutions for ODEs
We show a strange property of an approximate solution for the initial value problem of ODEs. This study is originally motivated by the numerical verification methods of solutions for parabolic
An improved method for verifying the existence and bounds of the inverse of second-order linear elliptic operators mapping to dual space
This paper presents an improved method for determining the invertibility of second-order linear elliptic operators with a bound on the norm of their inverses by computers in a mathematically rigorous
Numerical verification method of solutions for nonlinear elliptic and evolutional problems (Mathematical Analysis of Viscous Incompressible Fluid : RIMS研究集会報告集)
We consider the methods for guaranteed computations of solutions for nonlinear parabolic initial-boundary value problems. First, in order to make the basic principle clear, we briefly introduce the
A Framework for the Numerical Computation and A Posteriori Verification of Invariant Objects of Evolution Equations
TLDR
A theoretical framework for computer-assisted proofs of the existence of invariant objects in semilinear PDEs, including equilibrium points, traveling waves, periodic orbits and invariant manifolds attached to fixed points or periodic orbits is developed.
Numerical verification methods for a system of elliptic PDEs, and their software library
TLDR
Existing verification methods are reformulated using a convergence theorem for simplified Newton-like methods in the direct product space of a computable finite-dimensional space Vh and its orthogonal complement space V⊥ to provide verification methods of solutions to PDEs.
...
...

References

SHOWING 1-10 OF 15 REFERENCES
On the L 2 a Priori Error Estimates to the Finite Element Solution of Elliptic Problems with Singular Adjoint Operator
The Aubin–Nitsche trick for the finite element method of Dirichlet boundary value problem is a well-known technique to obtain a higher order a priori L 2 error estimation than that of estimates by
Computer-assisted proofs for semilinear elliptic boundary value problems
For second-order semilinear elliptic boundary value problems on bounded or unbounded domains, a general computer-assisted method for proving the existence of a solution in a “close” and explicit
A Numerical Method to Verify the Invertibility of Linear Elliptic Operators with Applications to Nonlinear Problems
TLDR
A numerical method to verify the invertibility of second-order linear elliptic operators by using the projection and the constructive a priori error estimates based upon the existing verification method originally developed by one of the authors is proposed.
On the Best Constant in the Error Bound for theH10-Projection into Piecewise Polynomial Spaces
Explicita priorierror bounds for the approximation by theH10-projection into piecewise polynomial spaces are given. In particular, for the quadratic approximation, the optimal constant is derived,
INTLAB - INTerval LABoratory
TLDR
INTLAB is a toolbox for Matlab supporting real and complex intervals, and vectors, full matrices and sparse matrices over those, designed to be very fast and achieves the anticipated speed with identical code on a variety of computers.
INTLAB–INTerval LABoratory, in Developments
  • Kyoto University E-mail address: kinosita@kurims.kyoto-u.ac.jp Sasebo National College of Technology,
  • 1999
...
...