We consider a general linear parabolic problem with extended time boundary conditions (including initial value problems and periodic ones), and approximate it by the implicit Euler scheme in time and the Gradient Discretisation method in space; the latter is in fact a class of methods that includes conforming and nonconforming finite elements, discontinuous Galerkin methods and several others. The main result is an error estimate which holds without supplementary regularity hypothesis on the solution. This result states that the approximation error has the same order as the sum of the interpolation error and the conformity error. The proof of this result relies on an inf-sup inequality in Hilbert spaces which can be used both in the continuous and the discrete frameworks. The error estimate result is illustrated by numerical examples with low regularity of the solution.

翻译：我们考虑一类带有扩展时间边界条件（包括初值问题和周期问题）的线性抛物问题，采用时间方向隐式欧拉格式和空间方向梯度离散法进行近似；后者实际上是一类包含协调与非协调有限元、间断伽辽金方法以及其他多种方法的统一框架。主要结果是一个无需对解附加正则性假设的误差估计。该结果指出：近似误差的阶数与插值误差及协调性误差之和的阶数一致。该证明依赖于希尔伯特空间中的inf-sup不等式，该不等式可同时应用于连续和离散框架。最后通过低正则性解的数值算例验证了该误差估计结果的有效性。