We analyze a new framework for expressing finite element methods on arbitrarily many intersecting meshes: multimesh finite element methods. The multimesh finite element method, first presented in [40], enables the use of separate meshes to discretize parts of a computational domain that are naturally separate; such as the components of an engine, the domains of a multiphysics problem, or solid bodies interacting under the influence of forces from surrounding fluids or other physical fields. Furthermore, each of these meshes may have its own mesh parameter. In the present paper we study the Poisson equation and show that the proposed formulation is stable without assumptions on the relative sizes of the mesh parameters. In particular, we prove optimal order a priori error estimates as well as optimal order estimates of the condition number. Throughout the analysis, we trace the dependence of the number of intersecting meshes. Numerical examples are included to illustrate the stability of the method.
翻译:我们分析了一个在任意的多相交的中间线上表达有限元素方法的新框架:多模点有限元素方法。多模点有限元素方法首先在 [40] 中首次提出,允许使用单独的中间线将自然分离的计算域部分分离;例如引擎的部件、多物理问题领域,或由周围液体或其他物理场的力量影响而相互作用的固体身体。此外,这些中间线中的每一种都可能有其自身的网状参数。在本文中,我们研究了Poisson方程,并表明拟议的配方是稳定的,没有假定网状参数的相对大小。特别是,我们证明最理想地订购了先验误估计数和条件号的最佳顺序估计。在分析过程中,我们追踪了相交的 meshes数量的依赖性。数字示例包括说明方法的稳定性。