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：我们采用连续Galerkin有限元方法提出了对流-扩散方程的全时空数值解。通过Galerkin/最小二乘法确保离散变分问题的稳定性。在全时空公式中，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。同时推导了后验误差估计，结合自适应时空网格细化提供了高效精确的数值解。通过解析解和传统时间推进算法的数值解对比，验证了时空解的精度。