Abstract We consider time discretization methods for abstract parabolic problems with inhomogeneous linear constraints. Prototype examples that fit into the general framework are the heat equation with inhomogeneous (time-dependent) Dirichlet boundary conditions and the time-dependent Stokes equation with an inhomogeneous divergence constraint. Two common ways of treating such linear constraints, namely explicit or implicit (via Lagrange multipliers) are studied. These different treatments lead to different variational formulations of the parabolic problem. For these formulations we introduce a modification of the standard discontinuous Galerkin (DG) time discretization method in which an appropriate projection is used in the discretization of the constraint. For these discretizations (optimal) error bounds, including superconvergence results, are derived. Discretization error bounds for the Lagrange multiplier are presented. Results of experiments confirm the theoretically predicted optimal convergence rates and show that without the modification the (standard) DG method has sub-optimal convergence behavior.

中文翻译：

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线性约束抛物线问题的不连续伽辽金时间离散化方法

摘要 我们考虑具有非齐次线性约束的抽象抛物线问题的时间离散化方法。适合一般框架的原型示例是具有非齐次（时间相关）狄利克雷边界条件的热方程和具有非齐次发散约束的时间相关斯托克斯方程。研究了处理此类线性约束的两种常见方法，即显式或隐式（通过拉格朗日乘子）。这些不同的处理导致抛物线问题的不同变分公式。对于这些公式，我们引入了标准不连续伽辽金 (DG) 时间离散化方法的修改，其中在约束离散化中使用了适当的投影。对于这些离散化（最佳）误差范围，包括超收敛结果，是派生的。提供了拉格朗日乘数的离散化误差界限。实验结果证实了理论上预测的最佳收敛速度，并表明（标准）DG 方法在没有修改的情况下具有次优收敛行为。