We introduce a new numerical method for the time-dependent Maxwell equations on unstructured meshes in two space dimensions. This relies on the introduction of a new mesh, which is the barycentric-dual cellular complex of the starting simplicial mesh, and on approximating two unknown fields with integral quantities on geometric entities of the two dual complexes. A careful choice of basis functions yields cheaply invertible block-diagonal system matrices for the discrete time-stepping scheme. The main novelty of the present contribution lies in incorporating arbitrary polynomial degree in the approximating functional spaces, defined through a new reference cell. The presented method, albeit a kind of Discontinuous Galerkin approach, requires neither the introduction of user-tuned penalty parameters for the tangential jump of the fields, nor numerical dissipation to achieve stability. In fact an exact electromagnetic energy conservation law for the semi-discrete scheme is proved and it is shown on several numerical tests that the resulting algorithm provides spurious-free solutions with the expected order of convergence.

我们为二维空间中非结构化网格上的时间相关麦克斯韦方程组引入了一种新的数值方法。这依赖于引入新的网格，该网格是开始的简单网格的重心-双细胞复合体，并且依赖于两个对偶复合体的几何实体上具有积分量的两个未知场的近似值。对于离散的时间步长方案，仔细选择基函数会产生便宜的可逆块对角线系统矩阵。本贡献的主要新颖之处在于将任意多项式并入通过新参考单元定义的近似函数空间中。所提出的方法虽然是一种不连续的Galerkin方法，但不需要为字段的切向跳跃引入用户调整的惩罚参数，也不能通过数值耗散来获得稳定性。实际上，已经证明了半离散方案的精确电磁能量守恒定律，并且在几个数值测试中表明，所得算法提供了具有预期收敛阶数的无杂散解。