In this article, we combine the local projection stabilization (LPS) technique in space and the discontinuous Galerkin (DG) method in time to investigate the time-dependent convection-diffusion-reaction problems. This kind of stabilized space-time finite element (STFE) scheme, based on approximation space enriched by bubble functions that can increase stability, is constructed. The existence, uniqueness and stability are proved with the properties of Lagrange interpolation polynomials established on Radau points in time direction. An error estimate in $L^{\infty }\left(L^2\right)$-norm is given by introducing the elliptic projection operators in space direction. This estimate approach is different from the previous ones that construct a special interpolant into approximation space showing an extra orthogonality property on the projection space. Since the techniques of Lagrange interpolation in time direction decouple time and space variables, the method proposed in this paper has the advantages of reducing calculation and simplifying theoretical analysis. The space and time convergence orders are illustrated in the first numerical example with smooth solutions. A comparison between the traditional STFE scheme and the constructed scheme for the problem having exponential boundary layers is presented in the second numerical example. The simulation results show that the novel method can greatly reduce nonphysical oscillations. And the influences of the stabilization parameters on the behavior of the approximate solution are discussed by some numerical results.

在本文中，我们将空间中的局部投影稳定（LPS）技术和不连续的Galerkin（DG）方法及时结合起来，研究了与时间有关的对流扩散反应问题。构造了这种稳定的时空有限元（STFE）方案，该方案基于具有可增加稳定性的气泡函数的近似空间。利用在时间方向上在Radau点上建立的Lagrange插值多项式的性质证明了其存在性，唯一性和稳定性。中的错误估计${\mathrm{大号}}^{\infty }（{\mathrm{大号}}^2）$-范数是通过在空间方向引入椭圆投影算子来给出的。这种估计方法不同于以前的方法，后者在近似空间中构造了一个特殊的插值，从而在投影空间上显示出额外的正交性。由于时间方向上的拉格朗日插值技术使时间和空间变量解耦，因此本文提出的方法具有减少计算量和简化理论分析的优点。在第一个数值示例中使用平滑解说明了空间和时间收敛阶数。在第二个数值示例中，对传统的STFE方案和构造的方案进行了比较，以解决具有指数边界层的问题。仿真结果表明，该方法可以大大减少非物理振荡。