A method for improving the accuracy of finite‐element solutions using the extrapolation method

To improve the accuracy of solutions by the finite‐element method (FEN), it is necessary to refine the element division or use higher‐order elements. However, in both approaches, large‐scale/long‐time computation is necessary because the dimension of the matrix equation to be solved increases. Especially, in vectorial or 3‐dimensional analysis, this is a serious problem with respect to computational cost. By the error analysis theory of FEM, the asymptotic property of errors is known when the interpolation polynomial of certain order is used. In light of this, we use the so‐called deferred approach to the limit and derive the extrapolation formula making use of the asymptotic property of errors. This leads to the more efficient use of FEM, which is one of our objectives. Also, an algorithm using the iterative extrapolation method is proposed to obtain more accurate solutions. The theory is tested numerically for the eigenvalue problem of a dielectric‐loaded rectangular waveguide. The estimate of the exact eigenvalue is obtained by using the extrapolation formula described above. Hence, it is shown that very accurate results are obtained by merely repeating small‐scale computations.