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A Numerical Method to Verify the Invertibility of Linear Elliptic Operators with Applications to Nonlinear Problems
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.
Verified numerical computation of solutions for the stationary Navier-Stokes equation in nonconvex polygonal domains
We propose a method to enclose solutions for the stationary Navier-Stokes equation in nonconvex polygonal domains. Our method is based on an infinite dimensional Newton-type formulation by using the…
Numerical validation of solutions of saddle point matrix equations
A numerical validation method for verifying the accuracy of approximate solutions of saddle point matrix equations is presented and it is shown that preconditioning can be used to improve the error bounds.
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…
Guaranteed error bounds for finite element approximations of noncoercive elliptic problems and their applications
Numerical verification of stationary solutions for Navier-Stokes problems
A numerical verification method for solutions of nonlinear parabolic problems
By using the finite element approximation and constructive a priori error estimates, a new formulation for proving the existence of solutions for nonlinear parabolic problems is presented. We present…
Constructive error analysis of a full-discrete finite element method for the heat equation
- K. Hashimoto, Takuma Kimura, Teruya Minamoto, M. Nakao
- Computer Science, MathematicsJapan Journal of Industrial and Applied…
- 19 June 2018
This work presents a new full-discrete finite element method for the heat equation, and shows the numerical stability of the method by verified computations, and effectively uses the constructive error estimates to effectively use the numerical computations with guaranteed accuracy.
Numerical enclosure of solutions for two dimensional driven cavity problems
The infinite dimensional Newton method takes an important role in this method, which needs to estimate the rigourous bound for the norm of inverse of the linearized operator.
A numerical verification method for solutions of singularly perturbed problems with nonlinearity
In order to verify the solutions of nonlinear boundary value problems by Nakao’s computerassisted numerical method, it is required to find a constant, as sharp as possible, in the a priori error…