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A posteriori error estimates of Darcy flows with Robin-type jump interface conditions

Jeonghun J. Lee

TL;DR

The paper develops a recovery-type a posteriori error estimator for dual mixed Darcy flow with Robin-type interface conditions across fractures, defined through Stenberg post-processing of the pressure. Reliability is proved using an interface-adapted Helmholtz decomposition and a specialized interpolation, with local efficiency and a path to bound the post-processed pressure by the estimator. The approach yields an energy-norm estimator η that guides adaptive mesh refinement, and numerical experiments confirm accurate error control and improved convergence, even under nonsmooth data and multiple fractures. This work advances reliable, fracture-aware adaptive methods for porous media flow without requiring a saturation assumption, and provides new analytic tools for interface-driven Darcy problems.

Abstract

In this work we develop an a posteriori error estimator for mixed finite element methods of Darcy flow problems with Robin-type jump interface conditions. We construct an energy-norm type a posteriori error estimator using the Stenberg post-processing. The reliability of the estimator is proved using an interface-adapted Helmholtz-type decomposition and an interface-adapted Scott--Zhang type interpolation operator. A local efficiency and the reliability of post-processed pressure are also proved. Numerical results illustrating adaptivity algorithms using our estimator are included.

A posteriori error estimates of Darcy flows with Robin-type jump interface conditions

TL;DR

The paper develops a recovery-type a posteriori error estimator for dual mixed Darcy flow with Robin-type interface conditions across fractures, defined through Stenberg post-processing of the pressure. Reliability is proved using an interface-adapted Helmholtz decomposition and a specialized interpolation, with local efficiency and a path to bound the post-processed pressure by the estimator. The approach yields an energy-norm estimator η that guides adaptive mesh refinement, and numerical experiments confirm accurate error control and improved convergence, even under nonsmooth data and multiple fractures. This work advances reliable, fracture-aware adaptive methods for porous media flow without requiring a saturation assumption, and provides new analytic tools for interface-driven Darcy problems.

Abstract

In this work we develop an a posteriori error estimator for mixed finite element methods of Darcy flow problems with Robin-type jump interface conditions. We construct an energy-norm type a posteriori error estimator using the Stenberg post-processing. The reliability of the estimator is proved using an interface-adapted Helmholtz-type decomposition and an interface-adapted Scott--Zhang type interpolation operator. A local efficiency and the reliability of post-processed pressure are also proved. Numerical results illustrating adaptivity algorithms using our estimator are included.
Paper Structure (12 sections, 10 theorems, 119 equations, 10 figures, 1 table)

This paper contains 12 sections, 10 theorems, 119 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries for governing equations
  3. Notation and definitions
  4. Governing equations and variational formulation
  5. Discretization with finite elements
  6. A posteriori error estimate
  7. Reliability estimate
  8. Local Efficiency
  9. Reliability of post-processed pressure
  10. Numerical results
  11. Conclusion
  12. Appendix: integration by parts identities

Key Result

Theorem 3.1

Suppose that $({\boldsymbol{u}}, p)$, $(\boldsymbol{u}_h, p_h)$ are solutions of eq:variational-eqs, eq:discrete-eqs, and $\eta$ is defined by eq:eta-def. Then, there exists $C>0$ independent of mesh sizes and $\alpha$ in eq:alpha-large such that

Figures (10)

  • Figure 1: A model domain with interface $\Gamma$
  • Figure 2: The domain with a vertical fault in numerical experiments (left figure) and the graphs of the pressure field in \ref{['eq:exact-pressure']} (middle and right figures).
  • Figure 3: Comparison of convergence of errors for uniform and adaptive refinements. The errors are computed with the manufactured solution \ref{['eq:exact-pressure']}. Pressure errors are computed with the post-processed pressure $p_h^*$.
  • Figure 4: The initial, and the 3rd, 6th, 9th refined meshes in adaptive solves with the manufactured solution \ref{['eq:exact-pressure']}.
  • Figure 5: Distribution of $\{\tilde{\eta}_{\Gamma,T}\}$ (left), $\{\tilde{\eta}_{0,T}\}$ (right) in the initial and the first mesh refinement for $\alpha = 0.1$ (color scale: white = 0, black = 2.0e-4)
  • ...and 5 more figures

Theorems & Definitions (21)

  • Theorem 3.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Theorem 3.2
  • ...and 11 more