<p style='text-indent:20px;'>Based on high order polynomial approximation and dimension reduction technique, we propose a novel numerical method for the fourth order Steklov problems in the circular domain. We first decompose the primal problem into a set of 1D problems via polar coordinate transformation and Fourier basis functions expansion. Then, by introducing a non-uniformly weighed Sobolev space, the variational form and corresponding discrete scheme are derived. Employing the Lax-Milgram lemma and approximation properties of the projection operators, we further prove existence and uniqueness of weak solutions and approximation solutions for each one-dimensional problems, and the error estimation between them, respectively. We also carry out ample numerical experiments which illustrate that the numerical algorithm is efficient and highly accurate.</p>
We present a high‐accuracy numerical method based on a decoupled dimensionality reduction scheme for Maxwell eigenvalue problem in spherical domains. Using the orthogonality of vector spherical harmonics and the variable separation approach, we decompose the original problem into two classes of decoupled one‐dimensional TE mode and TM mode. For the TE mode, we establish a variational formulation and its discrete scheme and give the error estimations of the approximate eigenvalues and eigenfunctions. For the TM mode, it is different from TE mode which naturally meets the divergence‐free condition and will not generate some spurious eigenvalues. We design a numerical algorithm based on a parameterized method to filter out the spurious eigenvalues. Finally, some numerical results are presented to confirm the theoretical results and validate the algorithms.
<abstract><p>We propose in this paper an efficient algorithm based on the Fourier spectral-Galerkin approximation for the fourth-order elliptic equation with periodic boundary conditions and variable coefficients. First, by using the Lax-Milgram theorem, we prove the existence and uniqueness of weak solution and its approximate solution. Then we define a high-dimensional $ L^2 $ projection operator and prove its approximation properties. Combined with Céa lemma, we further prove the error estimate of the approximate solution. In addition, from the Fourier basis function expansion and the properties of the tensor, we establish the equivalent matrix form based on tensor product for the discrete scheme. Finally, some numerical experiments are carried out to demonstrate the efficiency of the algorithm and correctness of the theoretical analysis.</p></abstract>
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