We present a range of mesh-dependent inequalities for piecewise constant and continuous piecewise linear finite element functions u defined on locally refined shape-regular (but possibly non-quasi-uniform) meshes. These inequalities involve norms of the form ∥h α u∥ W s,p (Ω) for positive and negative s and α, where h is a function which reflects the local mesh diameter in an appropriate way. The only global parameter involved is N, the total number of degrees of freedom in the finite element space, and we avoid estimates involving either the global maximum or minimum mesh diameter.\udOur inequalities include new variants of inverse inequalities as well as trace and extension theorems. They can be used in several areas of finite element analysis to extend results – previously known only for quasi-uniform meshes – to the locally refined case. Here we describe applications to (i) the theory of nonlinear approximation and (ii) the stability of the mortar element method for locally refined meshes.
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机译：我们为局部恒定形状和规则（但可能是非均匀）网格上定义的分段常数和连续分段线性有限元函数提供了一系列与网格相关的不等式。这些不等式涉及正负s和α的形式为∥hαu u W s，p（Ω）的范数，其中h是一个以适当方式反映局部网格直径的函数。所涉及的唯一全局参数是N，即有限元空间中的自由度总数，我们避免使用涉及全局最大或最小网格直径的估计。\ ud我们的不等式包括反不等式的新变体以及迹线和扩展定理。它们可用于有限元分析的多个领域，以将结果（以前仅对准均匀网格已知）扩展到局部精化的情况。在这里，我们描述了对（i）非线性近似理论和（ii）局部精炼网格的砂浆单元法的稳定性的应用。
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