Abstract Mixed finite element methods are applied to a Poisson problem with a singularity at a boundary point. The approximation spaces are based on quarter-point elements, the shape functions inheriting the singular behavior of their quadratic geometric maps. Two mesh scenarios are considered, by fixing some macro quarter-point elements at the coarse level, and subdividing them by mapping uniformly refined square meshes on the master element by their corresponding geometric transforms. For eight-noded coarse quadrilateral quarter-point elements, placing two mid-side nodes near the singular vertex, the radial singularity is exactly captured along element edges, and their refinements reveal shape regular curved meshes. For an improved version, using collapsed quadrilateral quarter-point elements obtained by reducing one of the quadrilateral element edges to the singular point, the radial singularity is captured inside the coarse macro elements as well. Their uniform refinement generates anisotropic meshes, grading towards the singular point. The assembly of the required H ( div ) -conforming approximation spaces based on these kinds of meshes are described. Results for a typical test problem demonstrate superior effectiveness of the proposed techniques for convergence acceleration, when confronted with usual affine finite elements, for h, p and hp-adaptive refinements. Especially, collapsed quarter-point elements applied to the singular problem reveal accuracy rates equivalent to standard regular contexts, of smooth solutions discretized on uniform affine meshes.

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基于一些各向异性和 hp 自适应曲线四分之一点元素的奇异椭圆问题的混合有限元逼近

摘要 将混合有限元方法应用于边界点为奇点的泊松问题。近似空间基于四分之一点元素，形状函数继承了其二次几何映射的奇异行为。考虑了两种网格场景，通过在粗略级别固定一些宏观四分之一点元素，并通过它们相应的几何变换在主元素上映射均匀细化的方形网格来细分它们。对于八节点粗四边形四分之一点单元，在奇异顶点附近放置两个中间节点，沿单元边缘精确捕获径向奇异点，并且它们的细化显示形状规则的弯曲网格。对于改进版本，使用通过将四边形元素边之一减少到奇异点而获得的折叠四边形四分之一点元素，径向奇异点也被捕获在粗宏元素内。它们的均匀细化生成各向异性网格，朝着奇异点渐变。描述了基于这些类型的网格所需的符合 H ( div ) 的近似空间的组装。典型测试问题的结果表明，当遇到通常的仿射有限元时，所提出的收敛加速技术对于 h、p 和 hp 自适应细化具有卓越的有效性。特别是，应用于奇异问题的折叠四分之一点元素显示的准确率相当于标准规则上下文，在均匀仿射网格上离散的平滑解。