标题： 一个产生完全可计算误差界限的拟插值算子 显示英文标题

标题： A quasi-interpolation operator yielding fully computable error bounds

摘要： 我们设计了一个从Sobolev空间$H^1_0(\Omega)$到由在单形网格上分片多项式形成的有限维有限元子空间的拟插值算子，该算子具有可计算的逼近常数。该算子1）在整个$H^1_0(\Omega)$上定义，不需要额外的正则性；2）允许任意多项式次数；3）适用于任何空间维度；4）在网格单元的顶点邻域内局部定义；5）对$H^1$半范数和$L^2$范数误差给出最优估计；6）对$H^1$半范数和$L^2$范数误差给出可计算的常数；7）导致全局最佳误差和局部最佳误差的等价性；8）具有投影性质。其构造遵循后验误差分析中的所谓势重构。 数值实验表明，我们的拟插值算子在$H^1$半范数和$L^2$范数中系统地给出了正确的收敛速率，其认证的过估计因子在所有测试情况下都非常精确且稳定。 显示英文摘要

摘要： We design a quasi-interpolation operator from the Sobolev space $H^1_0(\Omega)$ to its finite-dimensional finite element subspace formed by piecewise polynomials on a simplicial mesh with a computable approximation constant. The operator 1) is defined on the entire $H^1_0(\Omega)$, no additional regularity is needed; 2) allows for an arbitrary polynomial degree; 3) works in any space dimension; 4) is defined locally, in vertex patches of mesh elements; 5) yields optimal estimates for both the $H^1$ seminorm and the $L^2$ norm error; 6) gives a computable constant for both the $H^1$ seminorm and the $L^2$ norm error; 7) leads to the equivalence of global-best and local-best errors; 8) possesses the projection property. Its construction follows the so-called potential reconstruction from a posteriori error analysis. Numerical experiments illustrate that our quasi-interpolation operator systematically gives the correct convergence rates in both the $H^1$ seminorm and the $L^2$ norm and its certified overestimation factor is rather sharp and stable in all tested situations.