This article shows how to develop an efficient solver for a stabilized numerical space-time formulation of the advection-dominated diffusion transient equation. At the discrete space-time level, we approximate the solution by using higher-order continuous B-spline basis functions in its spatial and temporal dimensions. This problem is very difficult to solve numerically using the standard Galerkin finite element method due to artificial oscillations present when the advection term dominates the diffusion term. However, a first-order constraint least-square formulation allows us to obtain numerical solutions avoiding oscillations. The advantages of space-time formulations are the use of high-order methods and the feasibility of developing space-time mesh adaptive techniques on well-defined discrete problems. We develop a solver for a least-square formulation to obtain a stabilized and symmetric problem on finite element meshes. The computational cost of our solver is bounded by the cost of the inversion of the space-time mass and stiffness (with one value fixed at a point) matrices and the cost of the GMRES solver applied for the symmetric and positive definite problem. We illustrate our findings on an advection-dominated diffusion space-time model problem and present two numerical examples: one with isogeometric analysis discretizations and the second one with an adaptive space-time finite element method.

翻译：本文展示了如何为对流占优扩散瞬态方程的稳定数值时空格式开发一种高效求解算法。在离散时空层面，我们通过在其空间和时间维度上使用高阶连续B样条基函数来逼近解。由于当对流项主导扩散项时会产生人工振荡，该问题使用标准伽辽金有限元方法进行数值求解非常困难。然而，一阶约束最小二乘公式使我们能够获得避免振荡的数值解。时空格式的优势在于可以使用高阶方法，并且能够在定义良好的离散问题上开发时空网格自适应技术。我们为最小二乘公式开发了一种求解算法，以获得有限元网格上的稳定对称问题。我们求解器的计算成本受限于时空质量矩阵和刚度矩阵（其中一个值固定在某点）求逆的成本，以及应用于对称正定问题的GMRES求解器的成本。我们通过对流占优扩散时空模型问题阐明了我们的发现，并给出了两个数值示例：一个采用等几何分析离散化，另一个采用自适应时空有限元方法。