We present a framework for the structure-preserving approximation of partial differential equations on mapped multipatch domains, extending the classical theory of finite element exterior calculus (FEEC) to discrete de Rham sequences which are broken, i.e., fully discontinuous across the patch interfaces. Following the Conforming/Nonconforming Galerkin (CONGA) schemes developed in [http://dx.doi.org/10.1090/mcom/3079, arXiv:2109.02553, our approach is based on: (i) the identification of a conforming discrete de Rham sequence with stable commuting projection operators, (ii) the relaxation of the continuity constraints between patches, and (iii) the construction of conforming projections mapping back to the conforming subspaces, allowing to define discrete differentials on the broken sequence. This framework combines the advantages of conforming FEEC discretizations (e.g. commuting projections, discrete duality and Hodge-Helmholtz decompositions) with the data locality and implementation simplicity of interior penalty methods for discontinuous Galerkin discretizations. We apply it to several initial- and boundary-value problems, as well as eigenvalue problems arising in electromagnetics. In each case our formulations are shown to be well posed thanks to an appropriate stabilization of the jumps across the interfaces, and the solutions are extremely robust with respect to the stabilization parameter. Finally we describe a construction using tensor-product splines on mapped cartesian patches, and we detail the associated matrix operators. Our numerical experiments confirm the accuracy and stability of this discrete framework, and they allow us to verify that expected structure-preserving properties such as divergence or harmonic constraints are respected to floating-point accuracy.
翻译:我们提出了一个框架,用于在已绘制的多分流域上保存部分差异方程式的结构近似值,将限定元素外部微积分的传统理论(FEEC)扩展至断开的离散的Rham序列,即完全不连续地贯穿补丁界面。根据在[http://dx.doi.org/10.1090/mcom/3079,arXiv:2109.02553]中开发的 Conformation/不兼容的 Galerkin(CONGA) 方案,我们的方法基于:(一) 确定离散的 Rham 序列,与稳定的通勤预测操作操作员一致;(二) 放松补丁之间的连续性限制,以及(三) 构建符合匹配的预测图与符合的子空间,从而界定断开的序列上的离散差异。 这个框架将符合FEEC的离散化(e.g.commilutive预测, 离散的双向双向双向和Hodge-Helphotz decomposition) 依据:(一) 确定一个符合离散的离散的离散的离散的离散的离子和直径和直径直径精确的直径直径精确的离解的离解的离解和直径, 和直径流的直径直径直径直径直径直径直径直径直线的离解-直线的离解的离解的直线的直径, 。