拟合直边三角形网格上有限元在光滑边界上自然条件的场外强制
Optimal enforcement of natural conditions on smooth boundaries with piecewise polynomials upon fitted straight-edged meshes
AI总结：
研究因拟合直边三角形网格近似光滑域导致边界条件执行不准的问题，采用在双线性形式中添加项的方法，应用于三角形拉格朗日有限元，经数值实验验证，有效恢复精度并进行可靠性研究。
AI中文摘要：
几十年前，文献中讨论了由于拟合网格中直边三角形或四面体的并集对弯曲域的近似，导致在光滑域边界上规定的诺伊曼或罗宾条件执行不准确的一些可能补救措施。当时，如巴雷特和埃利奥特（1988）等作者主张使用具有单个弯曲边或面的单元，不仅在两三个顶点处，而且在这些曲线或曲面上的其他点处拟合真实边界，以定义与所用方法的理论近似阶兼容的某种类型的多项式曲面。在这项工作中，我们采用了一种不同的方法，其主要特点是仅使用由直边单元组成的拟合网格。通过在双线性形式中添加项来恢复由于多面体对域的近似而损失的精度，这些项考虑了要在近似边界上规定的相同类型的自然边界条件，尽管更接近真实条件。该技术在此应用于三角形拉格朗日有限元的情况，我们对反应扩散方程的解进行了严格的可靠性研究。提供了数值实验以支持理论结果。
英文摘要：
A few decades ago some possible remedies to an inaccurate enforcement of Neumann or Robin conditions prescribed on the boundary of a smooth domain, owing to its approximation by the union of straight-edged triangles or tetrahedra in a fitted mesh, were addressed. These studies, due to authors such as Barrett and Elliott (1988), advocated the use of isoparametric finite elements with a single curvilinear edge or face fitting the true boundary at two or three vertexes and also at additional points on those curves or curved surfaces, so as to define non polynomial piecewise solutions with the optimal order of approximation. In this work we adopt a different approach, whose main feature is the use of a fitted mesh consisting only of straight-edged elements, upon which the solution is a polynomial. Lost optimal orders of convergence, by virtue of the domain's approximation by a polytope, are recovered by means of the addition of terms to the bilinear form. These account for natural boundary conditions of the same type as the true ones, though to be prescribed on the approximating boundary instead. The new technique is applied to the case of triangular Lagrange finite elements, for which we give a rigorous reliability study in the solution of reaction-diffusion equations. Numerical experimentation is supplied in support of the theoretical results.
