  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.
