We consider fully discrete time-space approximations of abstract linear parabolic partial differential equations (PDEs) consisting of an $hp$-version discontinuous Galerkin (DG) time stepping scheme in conjunction with standard (conforming) Galerkin discretizations in space. We derive abstract computable a posteriori error bounds resulting, for instance, in concrete bounds in $L_{\infty}(I;L_2(\Omega))$- and $L_{2}(I;H^{1}(\Omega))$-type norms when $I$ is the temporal and $\Omega$ the spatial domain for the PDE. We base our methodology for the analysis on a novel space-time reconstruction approach. Our approach is flexible as it works for any type of elliptic error estimator and leaves their choice of up to the user. It also allows exhibits mesh-change estimators in a clear an concise way. We also show how our approach allows the derivation of such bounds in the $H^1(I;H^{-1}(\Omega))$ norm.
翻译:我们考虑完全离散的时间-空间近似点,即抽象线性抛物线部分差异方程式(PDEs)的完全离散的时间-空间近似点,该方方程式由美元转换为美元,Galerkin(DG)时间跨步制(DG)与空间标准(符合)加列金分离(Galerkin)时间跨步制(Galerkin)相配合。我们抽象地计算出一个事后误差的界限,例如,在L ⁇ infty}(I;L_2(\Omega)美元和$L ⁇ 2}(I;H ⁇ 1}(\(Omega))的具体界限中,结果为美元和美元(L ⁇ 2}(I;H ⁇ 1}(Omega))美元,而美元是PDE空间域的时值。我们把分析方法建立在新的空间-时间重建方法上。我们的方法是灵活的,因为它适用于任何种类的椭误算法,让用户选择。它还允许以简洁的方式展示模变估测度。我们的方法如何允许在$H_1(I);H ⁇ -1(H_)标准中得出这种界限。