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Perturbed saddle-point problems in $\mathbf{L}^p$ with non-regular loads

Abeer F. Alsohaim, Tomas Führer, Ricardo Ruiz-Baier, Segundo Villa-Fuentes

Abstract

In this work, we develop the discrete solvability analysis for perturbed saddle-point problems in Banach spaces with forcing terms regularised by means of a projector constructed using the adjoint of a weighted Clément quasi-interpolation. We take as driving example the linearised Poisson--Boltzmann (an advection-diffusion-reaction problem) in mixed form. We use perturbation arguments on the continuous and discrete levels and then derive a priori estimates that remain valid when the load that appears on the right-hand side of the "second" equation is in $\mathrm{H}^{-1}$. Further, we show a supercloseness result and {analyse convergence} of an adequate adaptation of Stenberg postprocessing for mixed advection equations with non-regular data. We provide numerical results that illustrate the convergence of the proposed scheme.

Perturbed saddle-point problems in $\mathbf{L}^p$ with non-regular loads

Abstract

In this work, we develop the discrete solvability analysis for perturbed saddle-point problems in Banach spaces with forcing terms regularised by means of a projector constructed using the adjoint of a weighted Clément quasi-interpolation. We take as driving example the linearised Poisson--Boltzmann (an advection-diffusion-reaction problem) in mixed form. We use perturbation arguments on the continuous and discrete levels and then derive a priori estimates that remain valid when the load that appears on the right-hand side of the "second" equation is in . Further, we show a supercloseness result and {analyse convergence} of an adequate adaptation of Stenberg postprocessing for mixed advection equations with non-regular data. We provide numerical results that illustrate the convergence of the proposed scheme.
Paper Structure (13 sections, 13 theorems, 140 equations, 3 figures, 3 tables)

This paper contains 13 sections, 13 theorems, 140 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Continuous setting
  3. Analysis of the primal formulation
  4. Notation and approximation spaces for mixed formulations
  5. The perturbed regularised primal problem
  6. Mixed formulation
  7. Discrete problem
  8. Error estimates
  9. Stenberg-type postprocessing
  10. Construction of $\mathcal{Q}_h$
  11. Adjoint problem and accuracy enhancement
  12. Numerical experiments
  13. Conclusion

Key Result

Proposition 2.1

There exists a constant $C>0$ such that for any $g\in \widetilde{\mathrm{H}}^{-1}(\Omega)$ and $\psi_D\in \mathrm{H}^{1/2}_{00}(\Gamma_D)$, Problem eq:PBstrong:primal admits a unique solution $\psi\in \mathrm{H}^1_D(\Omega)$ and

Figures (3)

  • Figure 6.1: Example 1. Approximate solutions (line integral contours and magnitude for the pseudo potential flux and double layer potential profile) for the lowest-order mixed finite element scheme, and postprocessed potential. We show results using smooth and rough loadings (top and bottom rows, respectively).
  • Figure 6.2: Example 2. Approximate solutions for the lowest-order mixed finite element scheme and postprocessed potential using the regularisation $\mathcal{Q}_h$.
  • Figure 6.3: Example 3. Approximation of pseudo potential flux and streamlines, double layer potential and postprocessed potential for the case of $U_0 = 0.25$ (top row) and $U_0 = 0.0025$ (middle row). The bottom panels show the approximate velocity field and streamlines for the high and low intensity, and a sample coarse mesh after 3 refinements.

Theorems & Definitions (27)

  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof
  • Remark 2.6
  • ...and 17 more