We present a full space-time numerical solution of the advection-diffusion equation using a continuous Galerkin finite element method on conforming meshes. The Galerkin/least-square method is employed to ensure stability of the discrete variational problem. In the full space-time formulation, time is considered another dimension, and the time derivative is interpreted as an additional advection term of the field variable. We derive a priori error estimates and illustrate spatio-temporal convergence with several numerical examples. We also derive a posteriori error estimates, which coupled with adaptive space-time mesh refinement provide efficient and accurate solutions. The accuracy of the space-time solutions is illustrated against analytical solutions as well as against numerical solutions using a conventional time-marching algorithm.

翻译：本文提出了一种在全时空域上求解对流-扩散方程的数值方法，该方法在保形网格上采用连续 Galerkin 有限元法。为确保离散变分问题的稳定性，采用了 Galerkin/最小二乘方法。在全时空表述中，时间被视为另一个维度，时间导数被解释为场变量的附加对流项。我们推导了先验误差估计，并通过多个数值算例展示了时空收敛性。同时，我们还推导了后验误差估计，结合自适应时空网格细化技术，可获得高效且精确的数值解。通过与解析解以及采用传统时间推进算法的数值解进行对比，验证了时空解法的精度。