[Paper Review] A sequential least squares method for elliptic equations in non-divergence form

Ruo Li, Fanyi Yang|arXiv (Cornell University)|Jun 10, 2019

Advanced Numerical Methods in Computational Mathematics16 references4 citations

TL;DR

This paper proposes a novel sequential least squares finite element method for solving second-order elliptic equations in non-divergence form by decoupling the solution process into two steps: first approximating the gradient in a piecewise irrotational polynomial space, then solving for the primary variable in a continuous Lagrange finite element space. The method achieves optimal convergence rates in both $L^2$ and energy norms and provides a natural a posteriori error estimator enabling effective adaptive refinement, validated through numerical experiments in 2D and 3D domains.

ABSTRACT

We develop a new least squares method for solving the second-order elliptic equations in non-divergence form. Two least-squares-type functionals are proposed for solving the equations in two steps. We first obtain a numerical approximation to the gradient in a piecewisely irrotational polynomial space. Then together with the numerical gradient, we seek a numerical solution of the primitive variable in continuous finite element space. The variational setting naturally provides a posteriori error which could be used in an adaptive refinement algorithm. The error estimates in $L^2$ norm and energy norms for both two unknowns are derived. By a series of numerical experiments, we verify the convergence rates and show the efficiency of the adaptive algorithm.

Motivation & Objective

• To develop a stable and convergent finite element method for second-order elliptic PDEs in non-divergence form, which lack a standard variational formulation due to the non-divergence structure.
• To overcome the challenges of $C^1$-conforming finite elements by using $C^0$ finite element spaces and avoiding saddle-point systems.
• To introduce a two-step sequential least squares approach that decouples gradient approximation from the primary variable solution.
• To derive a posteriori error estimators from the least squares functional for use in adaptive mesh refinement, especially for problems with low-regularity solutions.
• To validate the theoretical convergence rates and the efficiency of the adaptive algorithm through extensive numerical experiments in 2D and 3D.

Proposed method

• The method reformulates the non-divergence form PDE as a first-order system, enabling the use of least squares principles.
• It first minimizes a least squares functional over a piecewise irrotational polynomial space to approximate the gradient $\boldsymbol{p}$.
• Then, using the computed gradient, it solves a second least squares problem in a continuous $C^0$ finite element space for the primary variable $u$.
• The least squares functional naturally provides an a posteriori error estimator for adaptive refinement, based on the residual in the weak form.
• The method employs discontinuous Galerkin-type trace operators and jump terms to handle discontinuities across element boundaries.
• An $h$-adaptive refinement strategy is implemented using the a posteriori error estimator to enrich regions of low regularity, such as corner singularities.

Experimental results

Research questions

• RQ1 Can a sequential least squares finite element method be designed for non-divergence form elliptic PDEs without requiring $C^1$-conforming elements?
• RQ2 Does the use of a piecewise irrotational space for gradient approximation lead to optimal convergence rates in both $L^2$ and energy norms?
• RQ3 Can the least squares functional serve as an effective a posteriori error estimator for adaptive mesh refinement in problems with low-regularity solutions?
• RQ4 How do the convergence rates of the gradient and primary variable compare with theoretical predictions in numerical experiments?
• RQ5 Does adaptive refinement based on the a posteriori error estimator significantly improve efficiency for singular solutions?

Key findings

• The method achieves optimal convergence rates of order $m+1$ in the $L^2$ norm and $m$ in the energy norm for the gradient $\boldsymbol{p}$, and $m+1$ in $L^2$ and $m$ in energy norm for the primary variable $u$, matching theoretical predictions.
• Numerical experiments in 2D and 3D confirm the convergence orders for all error norms, with results aligning perfectly with theoretical estimates.
• For problems with corner singularities (e.g., $u = |\boldsymbol{x}|^{1.2}$), uniform refinement yields suboptimal convergence rates of $O(h^{0.2})$ for the gradient in the energy norm and $O(h^1)$ in $L^2$ norm due to reduced regularity.
• Adaptive refinement using the a posteriori error estimator leads to optimal convergence rates for all error norms, demonstrating superior efficiency over uniform refinement.
• The adaptive meshing strategy successfully concentrates refinement near singular regions, such as corners, as shown in the mesh refinement patterns after six steps.
• The method avoids solving saddle-point problems and only involves first-order operators in each step, simplifying implementation compared to mixed or mixed-hybrid formulations.