An Efficient Hermite Finite Element Method for One-dimensional Second-order Elliptic Interface Problems
Lu Zhang *
School of Mathematical Sciences, Guizhou Normal University, Guiyang 550025, China.
Zhongrong Xiang
School of Mathematical Sciences, Guizhou Normal University, Guiyang 550025, China.
Rong Huang
School of Mathematical Sciences, Guizhou Normal University, Guiyang 550025, China.
*Author to whom correspondence should be addressed.
Abstract
An efficient Hermite finite element method is proposed for one-dimensional second-order elliptic interface problems. Suitable Sobolev and finite element spaces are introduced, and the weak formulation together with its discrete counterpart is established. Existence and uniqueness of both the weak and discrete solutions are then proved by the Lax–Milgram lemma. The method employs piecewise cubic Hermite approximation on a fitted mesh aligned with the interface points, allowing discontinuous coefficients and interface conditions to be incorporated naturally through elementwise assembly. The proposed method has a relatively simple structure and avoids enriched approximation spaces. Numerical experiments confirm the accuracy and effectiveness of the proposed method.
Keywords: Elliptic interface problem, hermite finite element method, well-posedness of the solution, algorithm implementation, numerical experiment