An alternative approach to norm bound computation for inverses of linear operators in Hilbert spaces

@article{Kinoshita2019AnAA,
  title={An alternative approach to norm bound computation for inverses of linear operators in Hilbert spaces},
  author={Takehiko Kinoshita and Yoshitaka Watanabe and Mitsuhiro T. Nakao},
  journal={Journal of Differential Equations},
  year={2019}
}

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References

SHOWING 1-10 OF 50 REFERENCES

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

Some considerations of the invertibility verifications for linear elliptic operators

This paper presents three computer-assisted procedures for verifying the invertibility of second-order linear elliptic operators and for computing a bound on the norm of its inverse. One of these

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.

A Guaranteed Bound of the Optimal Constant in the Error Estimates for Linear Triangular Elements

TLDR
This paper provides the basic idea and outline the concept of verification procedures as well as show the final numerical result, which is a sufficiently sharp bound of the desired constant by a computer assisted proof.

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.

Numerical verification methods for solutions of semilinear elliptic boundary value problems

This article describes a survey on numerical verification methods for second-order semilinear elliptic boundary value problems introduced by authors and their colleagues. Here “numerical

Numerical Verification of Solutions for Nonlinear Elliptic Problems Using anL∞Residual Method☆

Abstract We consider a numerical enclosure method with guaranteedL∞error bounds for the solution of nonlinear elliptic problems of second order. By using an a posteriori error estimate for the