Startseite Mathematik Finite Element Penalty Method for the Oldroyd Model of Order One with Non-smooth Initial Data
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Finite Element Penalty Method for the Oldroyd Model of Order One with Non-smooth Initial Data

  • Bikram Bir , Deepjyoti Goswami ORCID logo und Amiya K. Pani ORCID logo EMAIL logo
Veröffentlicht/Copyright: 12. Februar 2022
Computational Methods in Applied Mathematics
Aus der Zeitschrift Computational Methods in Applied Mathematics Band 22 Heft 2

Abstract

In this paper, a penalty formulation is proposed and analyzed in both continuous and finite element setups, for the two-dimensional Oldroyd model of order one, when the initial velocity is in 𝐇 0 1 . New regularity results which are valid uniformly in time as t → ∞ and in the penalty parameter ε as ε → 0 are derived for the solution of the penalized problem. Then, based on conforming finite elements to discretize the spatial variables and keeping temporal variable continuous, a semidiscrete problem is discussed and a uniform-in-time a priori bound of the discrete velocity in Dirichlet norm is derived with the help of a penalized discrete Stokes operator and a modified uniform Gronwall’s lemma. Further, optimal error estimates for the penalized velocity in 𝐋 2 as well in 𝐇 1 -norms and for the penalized pressure in L 2 -norm have been established for the semidiscrete problem with non-smooth data. These error estimates hold uniformly in time under uniqueness assumption and also in the penalty parameter as it goes to zero. Our analysis relies on the suitable use of the inverse of the penalized Stokes operator, penalized Stokes–Volterra projection and judicious application of weighted time estimates with positivity property of the memory term. Finally, several numerical experiments are conducted on benchmark problems which confirm our theoretical findings.

Keywords: Oldroyd Fluid of Order One; Penalty Method; Uniform-in-Time Bound; Optimal Error Estimates; Non-smooth Initial Data
MSC 2010: 65M60; 65M15; 35Q35

Funding statement: The first author would like to express his gratitude to the Department of Science and Technology (DST), Government of India, for the financial support (DST/INSPIRE Fellowship/IF170401).

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Received: 2022-01-09
Accepted: 2022-01-09
Published Online: 2022-02-12
Published in Print: 2022-04-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. A General Error Estimate For Parabolic Variational Inequalities
  3. Functional A Posteriori Error Estimates for the Parabolic Obstacle Problem
  4. Multi-Scale Paraxial Models to Approximate Vlasov–Maxwell Equations
  5. Finite Element Penalty Method for the Oldroyd Model of Order One with Non-smooth Initial Data
  6. Quasi-Newton Iterative Solution of Non-Orthotropic Elliptic Problems in 3D with Boundary Nonlinearity
  7. Improving Regularization Techniques for Incompressible Fluid Flows via Defect Correction
  8. Fully Discrete Finite Element Approximation of the MHD Flow
  9. Sparse Data-Driven Quadrature Rules via ℓ p -Quasi-Norm Minimization
  10. Identification of Matrix Diffusion Coefficient in a Parabolic PDE
  11. A Finite Element Method for Two-Phase Flow with Material Viscous Interface
  12. On the Finite Element Approximation of Fourth-Order Singularly Perturbed Eigenvalue Problems
  13. Application of Fourier Truncation Method to Numerical Differentiation for Bivariate Functions
  14. Factorized Schemes for First and Second Order Evolution Equations with Fractional Powers of Operators
https://doi.org/10.1515/cmam-2022-0012
Schlagwörter für diesen Artikel
Oldroyd Fluid of Order One; Penalty Method; Uniform-in-Time Bound; Optimal Error Estimates; Non-smooth Initial Data

Artikel in diesem Heft

  1. Frontmatter
  2. A General Error Estimate For Parabolic Variational Inequalities
  3. Functional A Posteriori Error Estimates for the Parabolic Obstacle Problem
  4. Multi-Scale Paraxial Models to Approximate Vlasov–Maxwell Equations
  5. Finite Element Penalty Method for the Oldroyd Model of Order One with Non-smooth Initial Data
  6. Quasi-Newton Iterative Solution of Non-Orthotropic Elliptic Problems in 3D with Boundary Nonlinearity
  7. Improving Regularization Techniques for Incompressible Fluid Flows via Defect Correction
  8. Fully Discrete Finite Element Approximation of the MHD Flow
  9. Sparse Data-Driven Quadrature Rules via ℓ p -Quasi-Norm Minimization
  10. Identification of Matrix Diffusion Coefficient in a Parabolic PDE
  11. A Finite Element Method for Two-Phase Flow with Material Viscous Interface
  12. On the Finite Element Approximation of Fourth-Order Singularly Perturbed Eigenvalue Problems
  13. Application of Fourier Truncation Method to Numerical Differentiation for Bivariate Functions
  14. Factorized Schemes for First and Second Order Evolution Equations with Fractional Powers of Operators
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