• Abstract

Weak Galerkin finite element method is introduced for solving wave equation with interface on weak Galerkin finite element space $(\mathcal{P}_k(K), \mathcal{P}_{k−1}(∂K), [\mathcal{P}_{k−1}(K)]^2).$ Optimal order a priori error estimates for both space-discrete scheme and implicit fully discrete scheme are derived in $L^∞(L^2)$ norm. This method uses totally discontinuous functions in approximation space and allows the usage of finite element partitions consisting of general polygonal meshes. Finite element algorithm presented here can contribute to a variety of hyperbolic problems where physical domain consists of heterogeneous media.

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• Abstract

A linearized transformed $L1$ Galerkin finite element method (FEM) is presented for numerically solving the multi-dimensional time fractional Schrödinger equations. Unconditionally optimal error estimates of the fully-discrete scheme are proved. Such error estimates are obtained by combining a new discrete fractional Grönwall inequality, the corresponding Sobolev embedding theorems and some inverse inequalities. While the previous unconditional convergence results are usually obtained by using the temporal-spatial error spitting approaches. Numerical examples are presented to confirm the theoretical results.

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Solution and analysis of mathematical programming problems may be simplified when these problems are symmetric under appropriate linear transformations. In particular, a knowledge of the symmetries may help decrease the problem dimension, reduce the size of the search space by means of linear cuts. While the previous studies of symmetries in the mathematical programming usually dealt with permutations of coordinates of the solutions space, the present paper considers a larger group of invertible linear transformations. We study a special case of the quadratic programming problem, where the objective function and constraints are given by quadratic forms. We formulate conditions, which allow us to transform the original problem to a new system of coordinates, such that the symmetries may be sought only among orthogonal transformations. In particular, these conditions are satisfied if the sum of all matrices of quadratic forms, involved in the constraints, is a positive definite matrix. We describe the structure and some useful properties of the group of symmetries of the problem. Besides that, the methods of detection of such symmetries are outlined for different special cases as well as for the general case.

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Some new sixth-order compact finite difference schemes for Poisson/Helmholtz equations on rectangular domains in both two- and three-dimensions are developed and analyzed. Different from a few sixth-order compact finite difference schemes in the literature, the finite difference and weight coefficients of the new methods have analytic simple expressions. One of the new ideas is to use a weighted combination of the source term at staggered grid points which is important for grid points near the boundary and avoids partial derivatives of the source term. Furthermore, the new compact schemes are exact for 2D and 3D Poisson equations if the solution is a polynomial less than or equal to 6. The coefficient matrices of the new schemes are $M$-matrices for Helmholtz equations with wave number $K ≤ 0,$ which guarantee the discrete maximum principle and lead to the convergence of the new sixth-order compact schemes. Numerical examples in both 2D and 3D are presented to verify the effectiveness of the proposed schemes.

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The paper investigates the robustness and parallel scaling properties of a novel physical factorization preconditioner with algebraic multigrid subsolves in the iterative solution of a cell-centered finite volume discretization of the three-dimensional multi-group radiation diffusion equations. The key idea is to take advantage of a particular kind of block factorization of the resulting system matrix and approximate the left-hand block matrix selectively spurred by parallel processing considerations. The spectral property of the preconditioned matrix is then analyzed. The practical strategy is considered sequentially and in parallel. Finally, numerical results illustrate the numerical robustness, computational efficiency and parallel strong and weak scalabilities over the real-world structured and unstructured coupled problems, showing its competitiveness with many existing block preconditioners.

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Physics-Informed Neural Network (PINN) represents a new approach to solve Partial Differential Equations (PDEs). PINNs aim to solve PDEs by integrating governing equations and the initial/boundary conditions (I/BCs) into a loss function. However, the imbalance of the loss function caused by parameter settings usually makes it difficult for PINNs to converge, e.g. because they fall into local optima. In other words, the presence of balanced PDE loss, initial loss and boundary loss may be critical for the convergence. In addition, existing PINNs are not able to reveal the hidden errors caused by non-convergent boundaries and conduction errors caused by the PDE near the boundaries. Overall, these problems have made PINN-based methods of limited use on practical situations. In this paper, we propose a novel physics-informed neural network, i.e. an adaptive physics-informed neural network with a two-stage training process. Our algorithm adds spatio-temporal coefficient and PDE balance parameter to the loss function, and solve PDEs using a two-stage training process: pre-training and formal training. The pre-training step ensures the convergence of boundary loss, whereas the formal training process completes the solution of PDE by balancing various loss functions. In order to verify the performance of our method, we consider the imbalanced heat conduction and Helmholtz equations often appearing in practical situations. The Klein-Gordon equation, which is widely used to compare performance, reveals that our method is able to reduce the hidden errors. Experimental results confirm that our algorithm can effectively and accurately solve models with unbalanced loss function, hidden errors and conduction errors. The codes developed in this manuscript are publicy available at https://github.com/callmedrcom/ATPINN.

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We proposed absorbing interface conditions for the simulation of linear wave propagation on non-uniform meshes. Based on the superposition principle of second-order linear wave equations, we decompose the interface condition problem into two subproblems around the interface: for the first one the conventional artificial absorbing boundary conditions is applied, while for the second one, the local analytic solutions can be derived. The proposed interface conditions permit a two-way transmission of low-frequency waves across mesh interfaces which can be supported by both coarse and fine meshes, and perform a one-way absorption of high-frequency waves which can only be supported by fine meshes when they travel from fine mesh regions to coarse ones. Numerical examples are presented to illustrate the efficiency of the proposed absorbing interface conditions.

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Nanoscale heat transfer cannot be described by the classical Fourier law due to the very small dimension, and therefore, analyzing heat transfer in nanoscale is of crucial importance for the design and operation of nano-devices and the optimization of thermal processing of nano-materials. Recently, time-fractional dual-phase-lagging (DPL) equations with temperature jump boundary conditions have showed promising for analyzing the heat conduction in nanoscale. This article proposes a numerical algorithm with high spatial accuracy for solving the time-fractional dual-phase-lagging nano-heat conduction equation with temperature jump boundary conditions. To this end, we first develop a fourth-order accurate and unconditionally stable compact finite difference scheme for solving this time-fractional DPL model. We then present a fast numerical solver based on the divide-and-conquer strategy for the obtained finite difference scheme in order to reduce the huge computational work and storage. Finally, the algorithm is tested by two examples to verify the accuracy of the scheme and computational speed. And we apply the numerical algorithm for predicting the temperature rise in a nano-scale silicon thin film. Numerical results confirm that the present difference scheme provides ${\rm min}\{2−α, 2−β\}$ order accuracy in time and fourth-order accuracy in space, which coincides with the theoretical analysis. Results indicate that the mentioned time-fractional DPL model could be a tool for investigating the thermal analysis in a simple nanoscale semiconductor silicon device by choosing the suitable fractional order of Caputo derivative and the parameters in the model.

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Based on the rational system of Legendre rational functions, we construct two set of new interpolation basis functions on the unbounded intervals. Their explicit expressions are derived, and fast and stable algorithms are provided for computing the new basis functions. As applications, new rational collocation methods based on these new basis functions are proposed for solving various second-order differential equations on the unbounded domains. Numerical experiments illustrate that our new methods are more effective and stable than the existing collocation methods.

• General Summary