We propose an enriched Galerkin-characteristics finite element method for numerical solution of convection-dominated problems. The method uses the modified method of characteristics for the integration of the total derivative in time, combined with the finite element method for the spatial discretization on unstructured grids. The ${\mathrm{L}}^2$-projection method is implemented for the evaluation of numerical solutions by tracking the departure points from the integration points in each element. In the present study, a family of quadrature rules are used to enrich the approximation of integrals in the ${\mathrm{L}}^2$-projection method. The use of quadrature rules as an enrichment procedure allows for spatial discretizations on coarse fixed meshes and no need to introduce time-dependent enrichments. This procedure offers a very great advantage over the conventional Galerkin-characteristics finite element method since the same governing matrix representation can be used during the entire time steeping process. We also propose a multilevel adaptive procedure in the enriched Galerkin-characteristics finite element method by monitoring the gradient of the solution in the computational domain during its advection. In comparison with the traditional finite element analysis with h-, p- and hp-version refinements, the present approach is much simpler, more robust and efficient, and it yields more accurate solutions for a fixed number of degrees of freedom without refining the mesh. To examine the performance of the proposed method we solve several test examples for convection-diffusion problems. Comparison to the conventional Galerkin-characteristics finite element method is also carried out in the present work. The aim of such enriched method compared to the classical finite element method is to solve the time-dependent convection-dominated problems efficiently and with an appropriate level of accuracy.

我们提出了一种对流占优问题的数值解的丰富的Galerkin特征有限元方法。该方法使用改进的特征方法对总导数进行及时积分，并结合有限元方法对非结构化网格进行空间离散化。这${\mathrm{大号}}^{\mathrm{2个}}$-投影方法是通过跟踪每个元素中积分点的偏离点而实现的，用于评估数值解。在本研究中，使用正交规则族来丰富积分中的积分近似值。${\mathrm{大号}}^{\mathrm{2个}}$-投影方法。使用正交规则作为富集过程可以在粗糙的固定网格上进行空间离散化，而无需引入与时间有关的富集。与常规的Galerkin特征有限元方法相比，此过程具有很大的优势，因为可以在整个时间浸泡过程中使用相同的控制矩阵表示。通过在平流过程中监视计算域中溶液的梯度，我们还提出了丰富的Galerkin特征有限元方法中的多级自适应程序。与使用h-，p-和hp的传统有限元分析相比通过版本改进，本方法更加简单，更加健壮和高效，并且无需固定网格即可针对固定数量的自由度提供更准确的解决方案。为了检验所提出方法的性能，我们解决了对流扩散问题的几个测试示例。在目前的工作中，还与传统的Galerkin特征有限元方法进行了比较。与经典有限元方法相比，这种富集方法的目的是有效并以适当的精度水平解决与时间有关的对流占优的问题。