We provide an effective finite element method to solve the Schrödinger eigenvalue problem with an inverse potential on a spherical domain. To overcome the difficulties caused by the singularities of coefficients, we introduce spherical coordinate transformation and transfer the singularities from the interior of the domain to its boundary. Then by using orthogonal properties of spherical harmonic functions and variable separation technique we transform the original problem into a series of one-dimensional eigenvalue problems. We further introduce some suitable Sobolev spaces and derive the weak form and an efficient discrete scheme. Combining with the spectral theory of Babuška and Osborn for self-adjoint positive definite eigenvalue problems, we obtain error estimates of approximation eigenvalues and eigenvectors. Finally, we provide some numerical examples to show the efficiency and accuracy of the algorithm.

In this paper, we mainly study the high-order numerical scheme of right Caputo time fractional differential equations with uniform accuracy. Firstly, we construct the high-order finite difference method for the right Caputo fractional ordinary differential equations (FODEs) based on piecewise quadratic interpolation. The local truncation error of right Caputo FODEs is given, and the stability analysis of the right Caputo FODEs is proved in detail. Secondly, the time fractional partial differential equations (FPDEs) with right Caputo fractional derivative is studied by coupling the time-dependent high-order finite difference method and the spatial central second-order difference scheme. Finally, three numerical examples are used to verify that the convergence order of high-order numerical scheme is $ 3-\lambda $ in time with uniform accuracy.

In this paper, the time fractional diffusion equations optimal control problem is solved by $3-\alpha$ order with uniform accuracy scheme in time and finite element method (FEM) in space. For the state and adjoint state equation, the piecewise linear polynomials are used to make the space variables discrete, and obtain the semidiscrete scheme of the state and adjoint state. The priori error estimates for the semidiscrete scheme for state and adjoint state equation are established. Furthermore, the $3-\alpha$ order uniform accuracy scheme is used to make the time variable discrete in the semidiscrete scheme and construct the full discrete scheme for the control problems based on the first optimal condition and ‘first optimize, then discretize’ approach. The fully discrete scheme’s stability and truncation error are analyzed. Finally, two numerical examples are denoted to show that the theoretical analysis are correct.

In this paper, we present a high-order approximate solution with uniform accuracy for nonlinear 3D Volterra integral equations. This numerical scheme is constructed based on the three-dimensional block cubic Lagrangian interpolation method. At the same time, we give the local truncation error analysis of the numerical scheme based on Taylor’s theorem. Through theoretical analysis, we reach the conclusion that the optimal convergence order of this high-order numerical scheme is 4. Finally, we verify the effectiveness and applicability of the method through four numerical examples.

In this paper, a topology optimization algorithm for the mechanical-electrical coupling problem of periodic composite materials is studied. Firstly, the homogenization problem of the mechanical-electrical coupling topology optimization problem of periodic composite materials is established by the multi-scale asymptotic expansion method. Secondly, the topology optimization algorithm for the mechanical-electrical coupling problem of periodic composite materials is constructed by finite element method, solid isotropic material with penalisation method and homogenization method. Finally, numerical results show that the proposed algorithm is effective to calculate the optimal structure of the periodic composite cantilever beam under the influence of the mechanical-electrical coupling.