Mixed Finite Element (MFE) method is a robust numerical technique for solving elliptic and parabolic partial differential equations (PDEs). However, MFE can generate solutions with strong unphysical oscillations and/or large numerical diffusion for hyperbolic type PDEs. For its part, Discontinuous Galerkin (DG) finite element method is well adapted to solve hyperbolic systems and can accurately reproduce solutions involving sharp fronts. Therefore, the combination of DG and MFE is a good strategy for solving hyperbolic/parabolic problems such as advection – diffusion/dispersion equations. The classical formulation of the two methods is based on operator and time splitting allowing for separate solutions to advection with an explicit scheme and to dispersion with an implicit scheme. However, this kind of approach has the following drawbacks: () it lacks efficiency, as two systems with different unknowns are solved at each time step, () it induces errors generated by the splitting, () it can be CPU wise-expensive because of the CFL constraint, and () it cannot be employed for steady-state transport simulations.

中文翻译：

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对流-色散方程的完全隐式边/面中心不连续伽辽金/混合有限元格式

混合有限元 (MFE) 方法是一种用于求解椭圆和抛物型偏微分方程 (PDE) 的稳健数值技术。然而，MFE 可以为双曲型偏微分方程生成具有强烈非物理振荡和/或大数值扩散的解。就其本身而言，间断伽辽金 (DG) 有限元方法非常适合求解双曲系统，并且可以准确地重现涉及尖锐锋面的解。因此，DG 和 MFE 的结合是解决双曲/抛物线问题（例如平流-扩散/色散方程）的良好策略。这两种方法的经典公式基于算子和时间分割，允许使用显式方案单独解决平流问题，并使用隐式方案解决色散问题。然而，这种方法有以下缺点：（）它缺乏效率，因为在每个时间步求解两个具有不同未知数的系统，（）它会导致分裂产生的错误，（）它可能会占用CPU资源，因为CFL 约束，并且 () 它不能用于稳态传输模拟。