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A Sequential Least Squares Method for Poisson Equation using A Patch Reconstructed Space

Ruo Li, Fanyi Yang

TL;DR

The work tackles efficient numerical solution of the Poisson equation by introducing a sequential least-squares finite element method. It constructs a patch-reconstructed, piecewise irrotational space for the flux, enabling decoupling of the flux and pressure computations, and proves energy- and $L^2$-error estimates for both components. Numerical experiments across varied meshes and dimensions confirm the expected convergence rates, highlight the method’s robustness to geometry and singularities, and demonstrate substantial DOF reductions compared to standard DLS approaches. The approach offers a flexible, efficient framework for high-order Poisson solvers with clear practical benefits in accuracy and computational cost.

Abstract

We propose a new least squares finite element method to solve the Poisson equation. By using a piecewisely irrotational space to approximate the flux, we split the classical method into two sequential steps. The first step gives the approximation of flux in the new approximation space and the second step can use flexible approaches to give the pressure. The new approximation space for flux is constructed by patch reconstruction with one unknown per element consisting of piecewisely irrotational polynomials. The error estimates in the energy norm and $L^2$ norm are derived for the flux and the pressure. Numerical results verify the convergence order in error estimates, and demonstrate the flexibility and particularly the great efficiency of our method.

A Sequential Least Squares Method for Poisson Equation using A Patch Reconstructed Space

TL;DR

The work tackles efficient numerical solution of the Poisson equation by introducing a sequential least-squares finite element method. It constructs a patch-reconstructed, piecewise irrotational space for the flux, enabling decoupling of the flux and pressure computations, and proves energy- and -error estimates for both components. Numerical experiments across varied meshes and dimensions confirm the expected convergence rates, highlight the method’s robustness to geometry and singularities, and demonstrate substantial DOF reductions compared to standard DLS approaches. The approach offers a flexible, efficient framework for high-order Poisson solvers with clear practical benefits in accuracy and computational cost.

Abstract

We propose a new least squares finite element method to solve the Poisson equation. By using a piecewisely irrotational space to approximate the flux, we split the classical method into two sequential steps. The first step gives the approximation of flux in the new approximation space and the second step can use flexible approaches to give the pressure. The new approximation space for flux is constructed by patch reconstruction with one unknown per element consisting of piecewisely irrotational polynomials. The error estimates in the energy norm and norm are derived for the flux and the pressure. Numerical results verify the convergence order in error estimates, and demonstrate the flexibility and particularly the great efficiency of our method.

Paper Structure

This paper contains 13 sections, 9 theorems, 70 equations, 7 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Discontinuous Least Squares Finite Element Method
  3. Approximation Space with Irrotational Basis
  4. Sequential Least Squares Finite Element Approximation
  5. Numerical Results
  6. Convergence order study
  7. Example 1
  8. Example 2
  9. Example 3
  10. Example 4
  11. Example 5
  12. Efficiency comparison
  13. Conclusion

Key Result

Lemma 1

For any $(v_h, \bm{\mathrm{q}}_h) \in V_h^m \times \bm{\mathrm{W}}_h^m$, there exists a constant $C$ such that

Figures (7)

  • Figure 1: The triangular meshes with mesh size $h = 1/10$ (left) and $h = 1/20$ for Example 1.
  • Figure 2: The polygonal meshes with 250 elements (left) / 1000 elements (right).
  • Figure 3: The triangular meshes with 246 elements (left) and 984 elements (right) for Example 3.
  • Figure 4: The triangular meshes with 250 elements (left) and 1000 elements for Example 3.
  • Figure 5: Comparison of the error $\|\bm{\mathrm{p}} - \bm{\mathrm{p}}_h\|_{\bm{\mathrm{p}}}$ in number of DOFs by two methods with $m =1, 2, 3$ in two dimension (left) and three dimension (right).
  • ...and 2 more figures

Theorems & Definitions (17)

  • Lemma 1
  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 2
  • proof
  • Lemma 4
  • proof
  • ...and 7 more