Explicit a posteriori and a priori error estimation for the finite element solution of Stokes equations

@article{Liu2020ExplicitAP,
  title={Explicit a posteriori and a priori error estimation for the finite element solution of Stokes equations},
  author={Xuefeng Liu and Mitsuhiro T. Nakao and Chun'guang You and Shin'ichi Oishi},
  journal={ArXiv},
  year={2020},
  volume={abs/2006.02952}
}
For the Stokes equation over 2D and 3D domains, explicit a posteriori and a priori error estimation are novelly developed for the finite element solution. The difficulty in handling the divergence-free condition of the Stokes equation is solved by utilizing the extended hypercircle method along with the Scott-Vogelius finite element scheme. Since all terms in the error estimation have explicit values, by further applying the interval arithmetic and verified computing algorithms, the computed… 
1 Citations
Computer-assisted proof for the stationary solution existence of the Navier-Stokes equation over 3D domains
  • X. Liu, M. Nakao, S. Oishi
  • Computer Science, Mathematics
    Communications in Nonlinear Science and Numerical Simulation
  • 2022
TLDR
A computer-assisted solution existence verification method is proposed for the stationary Navier–Stokes equation over general 3D domains that verifies that the exact solution as the fixed point of the Newton iteration exists around the approximate solution.

References

SHOWING 1-10 OF 22 REFERENCES
A posteriori error estimators for the Stokes equations
SummaryWe present two a posteriori error estimators for the mini-element discretization of the Stokes equations. One is based on a suitable evaluation of the residual of the finite element solution.
A posteriori and constructive a priori error bounds for finite element solutions of the Stokes equations
We describe a method to estimate the guaranteed error bounds of the finite element solutions for the Stokes problem in mathematically rigorous sense. We show that an a posteriori error can be
On the divergence-free finite element method for the Stokes equations and the P 1 Powell-Sabin divergence-free element
A general framework for the conforming kk−1 mixed element method is set up. Because such a method would always produce point wise divergence-free solutions for the velocity and the mixed element can
Explicit lower bounds for Stokes eigenvalue problems by using nonconforming finite elements
An algorithm is proposed to give explicit lower bounds of the Stokes eigenvalues by utilizing two nonconforming finite element methods: Crouzeix–Raviart (CR) element and enriched Crouzeix–Raviart
A new family of stable mixed finite elements for the 3D Stokes equations
TLDR
This paper is devoted to proving that the mixed finite elements of this P k -P k-1 type when k > 3 satisfy the stability condition-the Babuska-Brezzi inequality on macro-tetrahedra meshes where each big tetrahedron is subdivided into four subtetrahedral meshes.
Remarks on a posteriori error estimation for finite element solutions
We utilize the classical hypercircle method and the lowest-order Raviart-Thomas H(div) element to obtain a posteriori error estimates of the P1 finite element solutions for 2D Poisson's equation. A
Verified Eigenvalue Evaluation for the Laplacian over Polygonal Domains of Arbitrary Shape
  • X. Liu, S. Oishi
  • Mathematics, Computer Science
    SIAM J. Numer. Anal.
  • 2013
TLDR
The proposed algorithm can provide concrete eigenvalue bounds for domain of arbitrary shape, even in the case that eigenfunction has singularity, when the finite element method is applied to bound leading eigenvalues of Laplace operator over polygonal domain.
Explicit Finite Element Error Estimates for Nonhomogeneous Neumann Problems
The paper develops an explicit a priori error estimate for finite element solution to nonhomogeneous Neumann problems. For this purpose, the hypercircle over finite element spaces is constructed and
A Numerical Verification Method of Solutions for the Navier-Stokes Equations
TLDR
A verification algorithm is presented which generates automatically on a computer a set including the exact solution of the stationary Navier-Stokes equations based on the infinite dimensional fixed point theorem using the Newton-like operator.
A framework of verified eigenvalue bounds for self-adjoint differential operators
  • X. Liu
  • Mathematics, Computer Science
    Appl. Math. Comput.
  • 2015
TLDR
For eigenvalue problems of self-adjoint differential operators, a universal framework is proposed to give explicit lower and upper bounds for the eigenvalues by applying Crouzeix-Raviart finite elements and adopting the interval arithmetic.
...
1
2
3
...