We consider goal-oriented adaptive space-time finite-element discretizations of the parabolic heat equation on completely unstructured simplicial space-time meshes. In some applications, we are interested in an accurate computation of some possibly nonlinear functionals at the solution, so called goal functionals. This motivates the use of adaptive mesh refinements driven by the dual-weighted residual (DWR) method. The DWR method requires the numerical solution of a linear adjoint problem that provides the sensitivities for the mesh refinement. This can be done by means of the same full space-time finite element discretization as used for the primal linear problem. The numerical experiment presented demonstrates that this goal-oriented, full space-time finite element solver efficiently provides accurate numerical results for a model problem with moving domains and a linear goal functional, where we know the exact value.

翻译：我们考虑在完全非结构化的单纯形时空网格上，对抛物型热方程进行面向目标的自适应时空有限元离散。在某些应用中，我们关心解的某些可能为非线性泛函的精确计算，即所谓的目标泛函。这促使我们采用由对偶加权残量方法驱动的自适应网格细化。对偶加权残量方法需要求解一个线性伴随问题，该问题提供网格细化的灵敏度信息。这可以通过与原始线性问题相同的全时空有限元离散来实现。所呈现的数值实验表明，对于具有移动域和线性目标泛函（已知精确值）的模型问题，这种面向目标的全时空有限元求解器能够高效地提供精确的数值结果。