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In this paper, we propose an effective finite element method for the fourth order elliptic eigenvalue problems in a circular domain. First, by using polar coordinates transformation and the orthogonal property of Fourier basis functions, the original problem is turned into a series of equivalent one-dimensional eigenvalue problems. Second, according to the properties of Laplace operator in polar coordinate, we deduce the polar conditions and introduce suitable weighted Sobolev space. Based on the polar conditions and weighted Sobolev space, we establish the weak form and the corresponding discrete form. Third, we prove the error estimates of approximation eigenvalues and eigenvectors by means of the spectral theory of compact operators for each one-dimensional eigenvalue system. Finally, we provide some numerical experiments to show the validity of the algorithm and the correctness of the theoretical results. [ABSTRACT FROM AUTHOR]

Fractional vegetation cover (FVC) has a significant role in indicating changes in ecosystems and is useful for simulating growth processes and modeling land surfaces. The fine-resolution FVC products represent detailed vegetation cover information within fine grids. However, the long revisit cycle of satellites with fine-resolution sensors and cloud contamination has resulted in poor spatial and temporal continuity. In this study, we propose to derive a spatially and temporally continuous FVC dataset by comparing multiple methods, including the data-fusion method (STARFM), curve-fitting reconstruction (S-G filtering), and deep learning prediction (Bi-LSTM). By combining Landsat and Sentinel-2 data, the integrated FVC was used to construct the initial input of fine-resolution FVC with gaps. The results showed that the FVC of gaps were estimated and time-series FVC was reconstructed. The Bi-LSTM method was the most effective and achieved the highest accuracy (R2 = 0.857), followed by the data-fusion method (R2 = 0.709) and curve-fitting method (R2 = 0.705), and the optimal time step was 3. The inclusion of relevant variables in the Bi-LSTM model, including LAI, albedo, and FAPAR derived from coarse-resolution products, further reduced the RMSE from 5.022 to 2.797. By applying the optimized Bi-LSTM model to Hubei Province, a time series 30 m FVC dataset was generated, characterized by a spatial and temporal continuity. In terms of the major vegetation types in Hubei (e.g., evergreen and deciduous forests, grass, and cropland), the seasonal trends as well as the spatial details were captured by the reconstructed 30 m FVC. It was concluded that the proposed method was applicable to reconstruct the time-series FVC over a large spatial scale, and the produced fine-resolution dataset can support the data needed by many Earth system science studies. [ABSTRACT FROM AUTHOR]

In this paper, we propose an efficient finite-element method based on dimension reduction scheme to solve the fourth-order elliptic equation in a circular domain. First of all, by using polar coordinate transformation and Fourier basis function expansion, the necessary pole conditions are derived and the original problem is decomposed into a series of equivalent one-dimensional fourth-order problems. Based on the pole conditions, the appropriate weighted Sobolev spaces are introduced and the error estimate of finite-element solution is proved for each one-dimensional fourth-order problem. Then we construct the cubic Hermite interpolation basis functions and derive the matrix form corresponding to the discrete scheme. Furthermore, the sparsity of mass matrix and stiffness matrix are illustrated. Finally, we provide some numerical experiments, and the numerical results show that our method is very effective. [ABSTRACT FROM AUTHOR]