Variational time discretization schemes are getting of increasing importance for the accurate numerical approximation of transient phenomena. The applicability and value of mixed finite element methods in space for simulating transport processes have been demonstrated in a wide class of works. We consider a family of continuous Galerkin–Petrov time discretization schemes that is combined with a mixed finite element approximation of the spatial variables. The existence and uniqueness of the semidiscrete approximation and of the fully discrete solution are established. For this, the Banach–Nečas–Babuška theorem is applied in a non-standard way. Error estimates with explicit rates of convergence are proved for the scalar and vector-valued variable. An optimal order estimate in space and time is proved by duality techniques for the scalar variable. The convergence rates are analyzed and illustrated by numerical experiments, also on stochastically perturbed meshes.

中文翻译：

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使用时间上的连续有限元和空间上的混合有限元对抛物线问题进行离散化的误差分析

变分时间离散化方案对于瞬态现象的精确数值逼近变得越来越重要。混合有限元方法在空间中模拟运输过程的适用性和价值已在大量工作中得到证明。我们考虑一系列与空间变量的混合有限元近似相结合的连续 Galerkin-Petrov 时间离散化方案。建立了半离散近似和完全离散解的存在唯一性。为此，以非标准方式应用 Banach-Nečas-Babuška 定理。对于标量和向量值变量，证明了具有显式收敛速度的误差估计。通过标量变量的对偶技术证明了空间和时间上的最优阶估计。