A Nonconformal Adaptive $h$-Refinement Method for 2-D Finite Element Analysis Based on Normalized Edge Basis Functions

In finite element adaptive techniques, the mesh quality generated by mesh refinement methods is of paramount importance. Based on normalized edge basis functions, we design new basis functions specifically for mesh interface edges containing hanging nodes. These basis functions are defined across the multiple triangular elements sharing the same interface edge. The developed basis functions are capable of preserving the continuity of the tangential components of the numerical fields across the interface between coarse and fine mesh regions. The nonconforming refinement method employed in this work, which is based on the proposed basis functions, can maintain the mesh quality and accelerate the adaptive process, without the need to introduce additional unknowns. By integrating the proposed technique with the dual-weighted residual method, we apply it to the adaptive process of time-harmonic electromagnetic problems, and make comparisons with the bisection method, in order to validate the effectiveness and superiority of the presented approach.