We present a space–time least-squares finite element method for the heat equation. It is based on residual minimization in $L^2$ norms in space–time of an equivalent first order system. This implies that (i) the resulting bilinear form is symmetric and coercive and hence any conforming discretization is uniformly stable, (ii) stiffness matrices are symmetric, positive definite, and sparse, (iii) we have a local a-posteriori error estimator for free. In particular, our approach features full space–time adaptivity. We also present a-priori error analysis on simplicial space–time meshes which are highly structured. Numerical results conclude this work.

中文翻译：

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抛物线方程的时空最小二乘有限元

我们提出了热方程的时空最小二乘有限元方法。它基于残差最小化${\mathrm{大号}}^{\mathrm{2个}}$等效一阶系统的时空规范。这意味着（i）生成的双线性形式是对称的和矫顽的，因此任何符合离散关系都是一致稳定的；（ii）刚度矩阵是对称的，正定的且稀疏的；（iii）对于自由。特别是，我们的方法具有完整的时空适应性。我们还介绍了高度结构化的简单时空网格的先验误差分析。数值结果总结了这项工作。