We propose a least-squares formulation for parabolic equations in the natural $L^2(0,T;V^*)\times H$ norm which avoids regularity assumptions on the data of the problem. For the abstract heat equation the resulting bilinear form then is symmetric, continuous, and coercive. This among other things paves the ground for classical space-time a priori and a posteriori Galerkin frameworks for the numerical approximation of the solution of the abstract heat equation. Moreover, the approach is applicable in e.g. optimal control problems with (parametrized) parabolic equations, and for certification of reduced basis methods with parabolic equations.

翻译：我们提出了一种基于自然$L^2(0,T;V^*)\times H$范数的抛物方程最小二乘公式，该公式避免了对问题数据的正则性假设。对于抽象热方程，由此得到的双线性形式是对称、连续且强制的。这为经典时空先验和后验Galerkin框架在抽象热方程数值近似求解中的应用奠定了基础。此外，该方法可应用于诸如含(参数化)抛物方程的最优控制问题，以及抛物方程降基方法的认证中。