The locally modified finite element method, which is introduced in [Frei, Richter: SINUM 52(2014), p. 2315-2334], is a simple fitted finite element method that is able to resolve weak discontinuities in interface problems. The method is based on a fixed structured coarse mesh, which is then refined into sub-elements to resolve an interior interface. In this work, we extend the locally modified finite element method {in two space dimensions} to second order using an isoparametric approach in the interface elements. Thereby we need to take care that the resulting curved edges do not lead to degenerate sub-elements. We prove optimal a priori error estimates in the $L^2$-norm and in a discrete energy norm. Finally, we present numerical examples to substantiate the theoretical findings.

翻译：局部修正有限元方法（由Frei和Richter于《SINUM》52(2014)第2315-2334页提出）是一种简单且能解析界面问题中弱间断性的拟合有限元方法。该方法基于固定的结构化粗网格，通过细分子单元来解析内部界面。本文在二维空间中采用等参方法将界面元素上的局部修正有限元方法扩展至二阶精度。为此需确保生成的曲边不会导致子单元退化。我们证明了$L^2$范数和离散能量范数下的最优先验误差估计。最后通过数值算例验证理论结果。