A posteriori error estimators for fourth order elliptic problems with concentrated loads

TL;DR

This work tackles the biharmonic equation with a concentrated load $\Delta^2 u = \delta_{\mathbf{x}_0}$ in a polygonal domain, where the Dirac delta induces low regularity and motivates adaptive mesh refinement. It develops two residual-based a posteriori estimators within a $C^0$ interior penalty method: a primal-based estimator directly from the model equation, and a projection-based estimator obtained by projecting the delta onto the discrete space and incorporating the projection error. The authors prove reliability and efficiency of both estimators, and implement an adaptive finite element algorithm that demonstrates robustness and quasi-optimal convergence across various boundary conditions and extensions to general fourth-order elliptic operators. The results provide practical tools for accurate plate bending simulations with singular loads, balancing accuracy and computational cost through targeted refinement near singularities. Overall, the paper advances adaptive methods for challenging fourth-order problems with nonstandard source terms and boundary conditions, with clear implications for engineering applications involving point loads and singularities.

Abstract

In this paper, we study two residual-based a posteriori error estimators for the $C^0$ interior penalty method in solving the biharmonic equation in a polygonal domain under a concentrated load. The first estimator is derived directly from the model equation without any post-processing technique. We rigorously prove the efficiency and reliability of the estimator by constructing bubble functions. Additionally, we extend this type of estimator to general fourth-order elliptic equations with various boundary conditions. The second estimator is based on projecting the Dirac delta function onto the discrete finite element space, allowing the application of a standard estimator. Notably, we additionally incorporate the projection error into the standard estimator. The efficiency and reliability of the estimator are also verified through rigorous analysis. We validate the performance of these a posteriori estimates within an adaptive algorithm and demonstrate their robustness and expected accuracy through extensive numerical examples.

Figures (18)

• Figure 1: Domain $\Omega$ with interior angle $\omega$ contains a singular point $\mathbf{x}_0$.
• Figure 2: $\alpha_0$ in terms of the largest interior angle $\omega$.
• Figure 3: (a) A $P_3$ Lagrange element. (b) A $\widetilde{P}_5$$C^1$ conforming macro element.
• Figure 4: Example \ref{['exam2']}: initial mesh and adaptive numerical solution.
• Figure 5: Example \ref{['exam2']}: adaptive meshes generated by $\eta$.

Theorems & Definitions (32)

• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 2.3: Continuity and coercivity RH2012
• Lemma 2.4: Trace inequality BS2008
• Lemma 2.5: Inverse inequality BS2008
• Lemma 2.6: Interpolation error estimate BGS2010
• Lemma 2.7
• Definition 3.1: GPJ2009
• Lemma 3.1