On multiwavelet-based finite-element method

A new approach of using multiwavelets in the finite-element method for electromagnetic-wave problems is presented for the first time. In this approach, the multiscalets are employed as the basis functions. Due to the smoothness, completeness, compact support, and interpolation property of the multiscalets, in terms of the basis function and its derivatives, fast convergence in approximating a function is achieved. The new basis functions are εC¹, i.e., the first derivatives of the bases are continuous on the connecting nodes. Thus, the divergence-free condition is satisfied at the end points. The multiscalets, along with their derivatives, are orthonormal in the discrete sampling nodes. Therefore, no coupled system of equations in terms of the function and its derivative is involved, resulting in a simple and efficient algorithm. Numerical results demonstrate the high efficiency and accuracy of the new method. For a partially loaded waveguide problem, we have achieved a factor of 16 in memory reduction and 435 in CPU speedup over the linear edge-element method.