The tangential-displacement normal-normal-stress (TDNNS) method is a finite element method for mixed elasticity. As the name suggests, the tangential component of the displacement vector as well as the normal-normal component of the stress are the degrees of freedom of the finite elements. The TDNNS method was shown to converge of optimal order, and to be robust with respect to shear and volume locking. However, the method is slightly nonconforming, and an analysis with respect to the natural norms of the arising spaces was still missing. We present a sound mathematical theory of the infinite dimensional problem using the space $${{\mathbf {H}}}(\mathbf {curl})$$H(curl) for the displacement. We define the space $${\underline{{\mathbf {H}}}}({\text {div}}\,\mathbf {{div}})$$H̲(divdiv) for the stresses and provide trace operators for the normal-normal stress. Moreover, the finite element problem is shown to be stable with respect to the $${{\mathbf {H}}}(\mathbf {curl})$$H(curl) and a discrete $${\underline{{\mathbf {H}}}}({\text {div}}\,\mathbf {{div}})$$H̲(divdiv) norm. A-priori error estimates of optimal order with respect to these norms are obtained. Beside providing a new analysis for the elasticity equation, the numerical techniques developed in this paper are a foundation for more complex models from structural mechanics such as Reissner Mindlin plate equations, see Pechstein and Schöberl (Numerische Mathematik 137(3):713–740, 2017).

中文翻译：

---

使用自然范数分析TDNNS方法

切向位移法向法向应力 (TDNNS) 方法是一种用于混合弹性的有限元方法。顾名思义，位移矢量的切向分量以及应力的法向分量是有限元的自由度。TDNNS 方法被证明以最优阶收敛，并且在剪切和体积锁定方面具有鲁棒性。但是，该方法略有不一致，并且仍然缺少对产生空间的自然范数的分析。我们使用空间 $${{\mathbf {H}}}(\mathbf {curl})$$H(curl) 作为位移，提出了无限维问题的合理数学理论。我们定义空间 $${\underline{{\mathbf {H}}}}({\text {div}}\, \mathbf {{div}})$$H̲(divdiv) 用于应力，并为法向-法向应力提供迹算子。此外，有限元问题在 $${{\mathbf {H}}}(\mathbf {curl})$$H(curl) 和离散 $${\underline{{\ mathbf {H}}}}({\text {div}}\,\mathbf {{div}})$$H̲(divdiv) 范数。获得关于这些范数的最优顺序的先验误差估计。除了为弹性方程提供新的分析之外，本文中开发的数值技术是结构力学更复杂模型的基础，例如 Reissner Mindlin 板方程，参见 Pechstein 和 Schöberl (Numerische Mathematik 137(3):713–740， 2017）。有限元问题在 $${{\mathbf {H}}}(\mathbf {curl})$$H(curl) 和离散 $${\underline{{\mathbf { H}}}}({\text {div}}\,\mathbf {{div}})$$H̲(divdiv) 范数。获得关于这些范数的最优顺序的先验误差估计。除了为弹性方程提供新的分析之外，本文中开发的数值技术是结构力学更复杂模型的基础，例如 Reissner Mindlin 板方程，参见 Pechstein 和 Schöberl (Numerische Mathematik 137(3):713–740， 2017）。有限元问题在 $${{\mathbf {H}}}(\mathbf {curl})$$H(curl) 和离散 $${\underline{{\mathbf { H}}}}({\text {div}}\,\mathbf {{div}})$$H̲(divdiv) 范数。获得关于这些范数的最优顺序的先验误差估计。除了为弹性方程提供新的分析之外，本文中开发的数值技术是结构力学更复杂模型的基础，例如 Reissner Mindlin 板方程，参见 Pechstein 和 Schöberl (Numerische Mathematik 137(3):713–740， 2017）。