We present a full space-time numerical solution of the advection-diffusion equation using a continuous Galerkin finite element method. 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：我们提出了一种使用连续伽辽金有限元方法求解平流-扩散方程的全时空数值解。通过伽辽金/最小二乘方法确保离散变分问题的稳定性。在全时空格式中，时间被视为另一个维度，时间导数被解释为场变量的附加平流项。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。此外，我们还推导了后验误差估计，将其与自适应时空网格细化相结合，能够提供高效且精确的解。通过解析解以及使用传统时间推进算法的数值解，验证了时空解的精度。