查看更多>>摘要：In this paper, we consider the thermoporoelasticity problem in heterogeneous and fractured media. The mathematical model is described by a coupled system of equations for pressure, temperature, and displacements. We apply a multiscale approach to reduce the size of the discrete system. We use a continuous finite element method and a Discrete Fracture Model (DFM) for fine grid approximation. For coarse grid approximation, we apply the Generalized Multiscale Finite Element Method (GMsFEM). The main idea of this method is to calculate multiscale basis functions by solving local spectral problems. We present numerical results for two-and three-dimensional model problems in heterogeneous and heterogeneous fractured media. We calculate relative errors between the reference fine grid solution and the multiscale solution for different numbers of multiscale basis functions. The results show that the proposed method can provide good accuracy with a few degrees of freedom.(C) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：In this paper, we firstly explore the special structure of the discretized linear systems from the spatial fractional diffusion equations. The coefficient matrices of the resulting discretized systems have a diagonal-plus-Toeplitz structure. Because the resulting Toeplitz matrix is symmetric positive definite (SPD), then we can employ the & UTau; matrix to approximate it. By making use of the piecewise interpolation polynomials, we propose a new approximate inverse preconditioner to handle the diagonal-plus-Toeplitz coefficient matrices. The tau matrix-based approximate inverse (TAI) preconditioning technique can be implemented very efficiently by using discrete sine transforms(DST). Theoretically, we have proved that the spectrum of the resulting preconditioned matrices are clustered around one. Thus, Krylov subspace methods with the proposed preconditioners converge very fast. To demonstrate the efficiency of the new preconditioners, numerical experiments are implemented. The numerical results show that with the proper interpolation node numbers, the performance of the tau-matrix based preconditioning technique is better than the other tested preconditioners. (C)& nbsp;2022 Elsevier B.V. All rights reserved.
查看更多>>摘要：In this paper, focusing on a distributed optimal control problem for the elliptic equations with integral control constraint, we propose efficient block diagonal preconditioners to solve the corresponding linearized algebraic system with finite element methods. We derive the first order necessary and sufficient optimality conditions for an integral constraint on the control, and divide the discretized optimal conditions into three matrix forms. Then block-diagonal preconditioners for the corresponding linear algebraic systems are constructed. With respect to both the mesh size and the regularization parameter, the robustness of our preconditioners is proved in detail. In fact, the condition numbers of the preconditioned matrices are a constant for different parameters. Meanwhile, based on the equivalent matrix forms, an algorithm is proposed for this kind of constrained optimal control problems. Numerical experiments are given to depict the efficiency of our proposed preconditioners. (C) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：In this work, we propose a preconditioned minimal residual (MINRES) method for a class of non-Hermitian block Toeplitz systems. Namely, considering an mn-by-mn non-Hermitian block Toeplitz matrix T-(n,T-m) with m-by -m commuting Hermitian blocks, we first premultiply it by a simple permutation matrix to obtain a Hermitian matrix and then construct a Hermitian positive definite block circulant preconditioner for the modified matrix. Under certain conditions, we show that the eigenvalues of the preconditioned matrix are clustered around +/- 1 when n is sufficiently large. Due to the Hermitian nature of the modified matrix, MINRES with our proposed preconditioner can achieve theoretically guaranteed superlinear convergence under suitable conditions. In addition, we provide several useful properties of block circulant matrices with commuting Hermitian blocks, including diagonalizability and symmetrization. A generalization of our result to the multilevel block case is also provided. We in particular indicate that our work can be applied to the all-at-once systems arising from solving evolutionary partial differential equations. Numerical examples are given to illustrate the effectiveness of our preconditioning strategy. (C)& nbsp;2021 Elsevier B.V. All rights reserved.
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