基于局部残差极小化的精确对称线性弹性自适应超收敛混合有限元方法
An adaptive superconvergent mixed finite element method for exactly symmetric linear elasticity based on local residual minimization
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中文总结 AI 辅助
该研究提出一种基于局部残差极小化的精确对称线性弹性自适应超收敛混合有限元方法，建立了含Lamé参数的后验误差估计，数值算例验证了理论结果。
中文摘要 AI 辅助
我们针对线性弹性问题的精确对称混合有限元离散，引入了一种后验误差估计子，该估计子将Stenberg型后处理方案重构为离散对偶范数下的局部残差极小化问题。该构造可对偶变量形式给出关联局部残差的Riesz代表元，且无需额外计算成本，我们基于此构建了后验误差指标。我们建立了可靠性估计，其中明确追踪了Lamé参数的依赖关系；在标准可压缩 regime下，建立了无需辅助泡函数的局部效率估计；还基于鲁棒稳定性性质和Oswald平均算子，建立了替代可靠性估计，其常数在λ→∞时保持有界；局部效率估计在两种 regime下均以相同的有界行为成立。值得注意的是，单一指标和单一比较范数适用于两种 regime，这与现有基于超圆的估计子形成对比，后者针对不可压缩极限需要不同的构造。数值算例包括具有已知奇异解的基准问题和无解析解的问题，验证了理论结果。
英文摘要
We introduce an a posteriori error estimator for exactly symmetric mixed finite element discretizations of linear elasticity, based on recasting a Stenberg-type postprocessing scheme as a local residual minimization problem in a discrete dual norm. This construction yields, as a dual variable and at no additional computational cost, a Riesz representative of the associated local residual, from which we build an a posteriori error indicator. We establish a reliability estimate, with the dependence on the Lamé parameters tracked explicitly, and a local efficiency estimate, without auxiliary bubble functions, in the standard compressible regime, as well as an alternative reliability estimate, based on a robust stability property and an Oswald averaging operator, with a constant that remains bounded as $λ\to\infty$; the local efficiency estimate holds, with the same bounded behavior, uniformly in both regimes. Notably, a single indicator and a single comparison norm serve both regimes, in contrast with existing hypercircle-based estimators, which require a distinct construction for the incompressible limit. Numerical examples, including a benchmark with a known singular solution and one without an analytical solution, validate the theoretical findings.
