Maxwell's equations are considered with transparent boundary conditions, for initial conditions and inhomogeneity having support in a bounded, not necessarily convex three-dimensional domain or in a collection of such domains. The numerical method only involves the interior domain and its boundary. The transparent boundary conditions are imposed via a time-dependent boundary integral operator that is shown to satisfy a coercivity property. The stability of the numerical method relies on this coercivity and on an anti-symmetric structure of the discretized equations that is inherited from a weak first-order formulation of the continuous equations. The method proposed here uses a discontinuous Galerkin method and the leapfrog scheme in the interior and is coupled to boundary elements and convolution quadrature on the boundary. The method is explicit in the interior and implicit on the boundary. Stability and convergence of the spatial semidiscretization are proven, and with a computationally simple stabilization term, this is also shown for the full discretization.
翻译:马克斯韦尔的方程式与透明的边界条件、初始条件和不相容性等式一起考虑,这些条件和不相容性在交界的、不一定是相交的三维域或这类域的集合中都有支持。数字方法仅涉及内域及其边界。透明的边界条件是通过一个有时间依赖的边界整体操作器强加的,该操作器显示符合腐蚀性特性。数字方法的稳定性取决于这种共和性和从连续方程式的薄弱一阶配方所继承的分离方程式的反对称结构。此处提议的方法使用不连续的加勒金法和内部的跳蛙法,并与边界的边界元素和相连接。该方法在内部是明确的,在边界上是隐含的。空间半分解的稳定性和趋同性得到了证明,并且用一个计算简单的稳定术语,这也显示完全分解。