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Numerical Analysis of a Coupled 3D-1D Transport Problem

Alyssa Taylor-LaPole, Uzochi Gideon, Beatrice Riviere, Duygu Vargun

Abstract

A finite element solution coupled with an interior penalty discontinuous Galerkin solution are defined for the approximation of the coupled 3D-1D solute transport problem. Under sufficient regularity for the weak solutions, optimal error bounds are derived for the 3D concentration and 1D concentration, that are optimal with respect to the time step size and the mesh sizes. Numerical results verify the theoretical results.

Numerical Analysis of a Coupled 3D-1D Transport Problem

Abstract

A finite element solution coupled with an interior penalty discontinuous Galerkin solution are defined for the approximation of the coupled 3D-1D solute transport problem. Under sufficient regularity for the weak solutions, optimal error bounds are derived for the 3D concentration and 1D concentration, that are optimal with respect to the time step size and the mesh sizes. Numerical results verify the theoretical results.
Paper Structure (13 sections, 3 theorems, 69 equations, 3 figures, 3 tables)

This paper contains 13 sections, 3 theorems, 69 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Model problem
  3. Numerical scheme
  4. Well-posedness
  5. Error analysis
  6. Numerical results
  7. Manufactured solution
  8. Diagonal line example
  9. Conclusion
  10. Derivation of the numerical scheme
  11. Proof of coercivity \ref{['eq:coercivity']}
  12. Proof of positivity \ref{['eq:bpositivity']}
  13. Bound for term $T_5$

Key Result

Lemma 1

For each $n\geq 1$, there exists a unique solution $(c_h^{n}, \hat{c}_h^{n})\in V_h^\Omega\times V_h^\Lambda$ satisfying eq:FEMscheme-eq:DGscheme. In addition, if $\tau \leq 1/2$, there is a constant $M>0$ independent of $h$ and $\tau$ such that

Figures (3)

  • Figure 1: Numerical prediction at $T=1$ of the manufactured solution: left figure shows both concentrations in the whole domain and the right figure shows the concentrations in the plane $z=1/2$.
  • Figure 2: Simulation results of a given concentration administered at the inlet of $\Lambda$, traveling to the outlet and diffusing into $\Omega$ over a time interval $[0,1]$. Case 1: constant radius and global $\gamma$, Case 2: smooth varying radius and global $\gamma$, and Case 3: smooth varying radius and piecewise constant $\gamma$.
  • Figure 3: Concentrations $c_1$ plotted along the diagonal vessel defined in Section \ref{['sec:diagline']} at times $0.0125, 0.005$ and $1$. The top rows shows concentrations at each time in the same scale, while the bottom row has a varying scale to highlight the differences between cases.

Theorems & Definitions (5)

  • Lemma 1
  • proof
  • Lemma 2
  • Theorem 3
  • proof