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Abstract
In this paper, a new computational technique is presented for the first time. In this method, physical differential equations are incorporated into interpolations of basic element in finite element methods. This is named physical spline finite element method (PSFEM). Theoretically, the physical spline interpolation introduces many new features. First, physical equations can be used in the interpolations to make the interpolations problem-associated. The algorithm converges much faster than any general interpolation while keeping the simplicity of the first order Lagrange interpolation. Second, the concept of basis functions may need to be re-examined. Thirdly, basis functions could be complex without simple geometric explanations. The applications to typical one-dimensional electromagnetic problems show the great improvements of the newly developed PSFEM on accuracy, convergence and stability. It can be extended to other applications. Extension to two- and three-dimensional problems is briefly discussed in the final section.