The Applications of Cross-Line Elements in Solving High-Order Partial Differential Equations Using Recurrence Formula

Free Element Method (FrEM) is a strong-form numerical method, which combines the superiority of meshfree methods (MFMs) and finite element methods (FEMs), and has been used to solve many engineering problems. However, the complexity of constructing higher-order schemes limits its application in solving high-order partial differential equations (PDEs). In this paper, cross-line element (CLE) is developed for solving high-order PDEs based on FrEM. Instead of using quadrilateral or hexahedral elements, only two lines and three lines are used in 2D and 3D CLE-FrEM, respectively. On each line, the Lagrange interpolation is employed to evaluate the derivatives toward curves. The paper applied the derivative operator recurrently on the lower-order derivatives in FrEM to obtain higher-order partial derivatives. The paper compared the numerical accuracy and the computational efficiency between CLE-FrEM and FLM for challenging PDEs, such as the Korteweg–de Vries equation and Kuramoto–Sivashinsky equation. The results indicate that the proposed method has higher efficiency than FLM with the same accuracy.