We consider the numerical solution of an abstract operator equation $Bu=f$ by using a least-squares approach. We assume that $B: X \to Y^*$ is an isomorphism, and that $A : Y \to Y^*$ implies a norm in $Y$, where $X$ and $Y$ are Hilbert spaces. The minimizer of the least-squares functional $\frac{1}{2} \, \| Bu-f \|_{A^{-1}}^2$, i.e., the solution of the operator equation, is then characterized by the gradient equation $Su=B^* A^{-1}f$ with an elliptic and self-adjoint operator $S:=B^* A^{-1} B : X \to X^*$. When introducing the adjoint $p = A^{-1}(f-Bu)$ we end up with a saddle point formulation to be solved numerically by using a mixed finite element method. Based on a discrete inf-sup stability condition we derive related a priori error estimates. While the adjoint $p$ is zero by construction, its approximation $p_h$ serves as a posteriori error indicator to drive an adaptive scheme when discretized appropriately. While this approach can be applied to rather general equations, here we consider second order linear partial differential equations, including the Poisson equation, the heat equation, and the wave equation, in order to demonstrate its potential, which allows to use almost arbitrary space-time finite element methods for the adaptive solution of time-dependent partial differential equations.

翻译：本文考虑通过最小二乘法求解抽象算子方程 $Bu=f$。假设 $B: X \to Y^*$ 是一个同构映射，且 $A : Y \to Y^*$ 在 $Y$ 中定义了一个范数，其中 $X$ 和 $Y$ 是希尔伯特空间。最小二乘泛函 $\frac{1}{2} \, \| Bu-f \|_{A^{-1}}^2$ 的极小化问题（即算子方程的解）可从梯度方程 $Su=B^* A^{-1}f$ 中描述，其中 $S:=B^* A^{-1} B : X \to X^*$ 是一个椭圆自伴算子。引入伴随变量 $p = A^{-1}(f-Bu)$ 后，我们得到一个鞍点形式，并通过混合有限元方法进行数值求解。基于离散inf-sup稳定性条件，我们推导了相应的先验误差估计。虽然伴随变量 $p$ 在构造上为零，但其近似解 $p_h$ 可作为后验误差指示器，在适当离散化时驱动自适应方案。该方法可应用于相当广泛的方程，本文以二阶线性偏微分方程为例——包括泊松方程、热传导方程和波动方程——来展示其潜力，从而允许对时变偏微分方程的自适应求解使用几乎任意的时空有限元方法。