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Scaling-robust built-in a posteriori error estimation for discontinuous least-squares finite element methods

Philipp Bringmann

TL;DR

This work develops a scaling-robust, built-in a posteriori error framework for discontinuous least-squares finite element methods on piecewise Sobolev spaces, using a domain-weight $c_ ext{Ω}$ and a side condition on interelement jumps. A fundamental LS principle with a Helmholtz decomposition yields equivalence between the LS functional and domain-scale independent norms, enabling reliable a priori and a posteriori analysis for discontinuous discretizations, including PW Raviart–Thomas spaces. The authors extend the theory to over-penalized LSFEMs via averaging operators, establishing a robust built-in estimator and quasi-best-approximation properties. Numerical experiments on scaled squares, anisotropic rectangles, and L-shaped domains demonstrate the necessity of proper weighting for optimal adaptive convergence across polynomial degrees, validating the practical impact of the proposed weighting and discretization choices.

Abstract

A convincing feature of least-squares finite element methods is the built-in a posteriori error estimator for any conforming discretization. In order to generalize this property to discontinuous finite element ansatz functions, this paper introduces a least-squares principle on piecewise Sobolev functions by the example of the Poisson model problem with mixed boundary conditions. It allows for fairly general discretizations including standard piecewise polynomial ansatz spaces on triangular and polygonal meshes. The presented scheme enforces the interelement continuity of the piecewise polynomials by additional least-squares residuals. A side condition on the normal jumps of the flux variable requires a vanishing integral mean and enables the penalization of the jump with the natural power of the mesh size in the least-squares functional. This avoids over-penalization with additional regularity assumptions on the exact solution as usually present in the literature on discontinuous LSFEM. The proof of the built-in a posteriori error estimation for the over-penalized scheme is presented as well. All results in this paper are robust with respect to the size of the domain guaranteed by a suitable weighting of the residuals in the least-squares functional. Numerical experiments illustrate the importance of the proposed weighting and exhibit optimal convergence rates of the adaptive mesh-refining algorithm for various polynomial degrees.

Scaling-robust built-in a posteriori error estimation for discontinuous least-squares finite element methods

TL;DR

This work develops a scaling-robust, built-in a posteriori error framework for discontinuous least-squares finite element methods on piecewise Sobolev spaces, using a domain-weight and a side condition on interelement jumps. A fundamental LS principle with a Helmholtz decomposition yields equivalence between the LS functional and domain-scale independent norms, enabling reliable a priori and a posteriori analysis for discontinuous discretizations, including PW Raviart–Thomas spaces. The authors extend the theory to over-penalized LSFEMs via averaging operators, establishing a robust built-in estimator and quasi-best-approximation properties. Numerical experiments on scaled squares, anisotropic rectangles, and L-shaped domains demonstrate the necessity of proper weighting for optimal adaptive convergence across polynomial degrees, validating the practical impact of the proposed weighting and discretization choices.

Abstract

A convincing feature of least-squares finite element methods is the built-in a posteriori error estimator for any conforming discretization. In order to generalize this property to discontinuous finite element ansatz functions, this paper introduces a least-squares principle on piecewise Sobolev functions by the example of the Poisson model problem with mixed boundary conditions. It allows for fairly general discretizations including standard piecewise polynomial ansatz spaces on triangular and polygonal meshes. The presented scheme enforces the interelement continuity of the piecewise polynomials by additional least-squares residuals. A side condition on the normal jumps of the flux variable requires a vanishing integral mean and enables the penalization of the jump with the natural power of the mesh size in the least-squares functional. This avoids over-penalization with additional regularity assumptions on the exact solution as usually present in the literature on discontinuous LSFEM. The proof of the built-in a posteriori error estimation for the over-penalized scheme is presented as well. All results in this paper are robust with respect to the size of the domain guaranteed by a suitable weighting of the residuals in the least-squares functional. Numerical experiments illustrate the importance of the proposed weighting and exhibit optimal convergence rates of the adaptive mesh-refining algorithm for various polynomial degrees.
Paper Structure (13 sections, 13 theorems, 73 equations, 11 figures)

This paper contains 13 sections, 13 theorems, 73 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Piecewise Sobolev spaces and jumps
  3. Least-squares principle for piecewise Sobolev spaces
  4. Computable jump terms for piecewise polynomials
  5. Discontinuous least-squares FEM
  6. Error analysis
  7. A posteriori error estimation for over-penalized discontinuous LSFEM
  8. Numerical experiments
  9. Scaled square example
  10. Anisotropic rectangle example
  11. L-shaped domain example
  12. Proof of Lemma \ref{['lem:quasi_interpolation']}
  13. Piecewise Raviart--Thomas basis with vanishing mean normal jump

Key Result

Lemma 2.1

Let $\sigma, \tau \in H(\mathop{\mathrm{div}}\nolimits, \mathcal{T})$ and $u, v \in H^1(\mathcal{T})$. If $\sigma \in H_{\textup{N}}(\mathop{\mathrm{div}}\nolimits, \Omega)$ and $u \in H^1_{\textup{D}}(\Omega)$, then $s_n(\sigma + \tau) = s_n(\tau)$, $s_t(\nabla u + \nabla_{\textup{pw}} v) = s_t(\na

Figures (11)

  • Figure 1: Edge patch $\omega_E$
  • Figure 2: Illustration of initial triangulations $\mathcal{T}_0$ of the scaled square domain, the anisotropic rectangle domain, and the scaled L-shaped domain.
  • Figure 3: Convergence history plot of the relative energy error $\Vert \nabla(u - u_{\textup{LS}}) \Vert_{L^2(\Omega)} / \Vert \nabla u \Vert_{L^2(\Omega)}$ under uniform refinement of the scaled square $\Omega_\ell$ from Subsection \ref{['sec:scaling']}. The experiments are carried out with the naturally penalized discontinuous LSFEM ($\alpha = 1$) for lowest-order ansatz spaces ($k = 0$) and different weighting factors $c_\Omega$.
  • Figure 4: Convergence history plot of the relative energy error $\Vert \nabla(u - u_{\textup{LS}}) \Vert_{L^2(\Omega)} / \Vert \nabla u \Vert_{L^2(\Omega)}$ under uniform refinement of the square $\Omega_\pi$ with edge length $\ell = \pi$ from Subsection \ref{['sec:scaling']}. The experiments are carried out with the naturally penalized discontinuous LSFEM ($\alpha = 1$) for different polynomial degrees $k$ and weighting factors $c_\Omega$.
  • Figure 5: Convergence history plot of the relative energy error $\Vert \nabla(u - u_{\textup{LS}}) \Vert_{L^2(\Omega)} / \Vert \nabla u \Vert_{L^2(\Omega)}$ under uniform refinement of the scaled square $\Omega_\ell$ from Subsection \ref{['sec:scaling']}. The experiments are carried out with the over-penalized discontinuous LSFEM ($\alpha = -1$) for lowest-order ansatz spaces ($k = 0$) and different weighting factors $c_\Omega$.
  • ...and 6 more figures

Theorems & Definitions (26)

  • Lemma 2.1: consistency of jump terms
  • Lemma 2.2: equivalence of norms
  • Theorem 3.1: fundamental equivalence
  • Lemma 3.2: Helmholtz decomposition for piecewise Sobolev functions
  • Remark 3.3: scaling-robust conforming LSFEM
  • Remark 3.4: three space dimensions
  • Lemma 4.1: equivalence of edge-oriented measures
  • Remark 4.2: generalization of upper bound
  • Remark 4.3: necessity of a side condition
  • Corollary 4.4
  • ...and 16 more