We introduce a Scott--Zhang type projection operator mapping to Lagrange elements for arbitrary polynomial order. In addition to the usual properties, this operator is compatible with duals of first order Sobolev spaces. More specifically, it is stable in the corresponding negative norms and allows for optimal rates of convergence. We discuss alternative operators with similar properties. As applications of the operator we prove interpolation error estimates for parabolic problems and smoothen rough right-hand sides in a least squares finite element method.
翻译:我们引入了Scott-Zhang型投影操作员对任意多元顺序的 Lagrange 元素的绘图。 除了通常的特性外,该操作员与Sobolev 空间第一顺序的双重特性相容, 更具体地说, 它在相应的负规范中保持稳定, 并允许最佳的趋同率 。 我们讨论类似属性的替代操作员 。 作为操作员的应用, 我们证明了对抛物线问题和平滑的右侧的内插错误估计, 并且用最小方形限定元素法 。