This paper proposes an efficient deflated preconditioned conjugate gradient (DPCG) solver for linear elastic fracture problems based on the extended finite element method (XFEM). In the deflation technique, the deflated matrix incorporates not only the standard rigid body motion but also additional vectors that correspond to the enrichment functions in XFEM, which can eliminate the low-frequency iterative errors significantly. Furthermore, a special crack-tip domain decomposition preconditioner is designed to eliminate the high-frequency iterative errors. These two strategies are combined multiplicatively to yield a robust DPCG solver for the highly ill-conditioned systems generated by XFEM. In numerical experiments, a simple and effective criterion for selecting deflation nodes is presented, and the efficiency of the proposed algorithm is validated through different crack problems.

In this paper, we consider the efficient simulations of dynamic crack propagations based on the Extended Finite Element Method (XFEM). For the time discretization, the Generalized-a method is adopted to instead of the commonly used Newmark method in engineering, and the non physical numerical oscillations can be reduced in the Generalized-a method by choosing appropriate parameters. Moreover, in order to accelerate the convergence rate of the linear system arising from XFEM, a special crack-tip domain decomposition preconditioning method is developed, in which the computational domain is decomposed into regular subdomains and crack tip subdomains. To construct the Schwarz preconditioners, the subproblems are solved exactly in the crack tip subdomains and inexactly in the regular subdomains by an incomplete LU factorization. When cracks propagate, only the subdomains around the crack tips are updated, and all the other regular subdomains remain unchanged, which can save the computational cost significantly. The numerical experiments verify that the proposed preconditioning algorithm works well for the simulations of dynamic crack propagations.

In this paper, a new block preconditioner is proposed for the saddle point problem arising from the Neumann boundary control problem. In order to deal with the singularity of the stiffness matrix, the saddle point problem is first extended to a new one by a regularization of the pure Neumann problem. Then after row permutations of the extended saddle point problem, a new block triangular preconditioner is constructed based on an approximation of the Schur complement. We analyze the eigenvalue properties of the preconditioned matrix and provide eigenvalue bounds. Numerical results illustrate the efficiency of the proposed preconditioning method.

This article focuses on designing efficient iteration algorithms for nonequilibrium three-temperature heat conduction equations, which are used to formulate the radiative energy transport problem. Based on the framework of relaxation iteration, we design a new accelerated iteration algorithm by reasonable approximation of the Jacobi matrix, according to the characteristics of the discrete scheme for the three-temperature equations. Adopting the iteration framework, we analyze the advantages and disadvantages of several iteration algorithms commonly used in practice and the new iteration algorithm. Finally, we compare the new iteration algorithm with some other iteration algorithms by solving several nonlinear models, and show that the new algorithm can achieve significant acceleration effect.

In this paper, a domain decomposition based preconditioning method is developed to accelerate the Krylov subspace method for solving the linear system arising from the extended finite element discretization of the dynamic crack problem. Based on the observation that the crack tip area has a significant impact on the convergence of the iterative method while the other mesh points do not, the finite element mesh is partitioned into two types: the regular subdomains and the crack tip subdomains. To construct the additive Schwarz preconditioner, the global matrix is partitioned into submatrices which are solved exactly in the crack tip subdomains by the LU factorization and inexactly in the regular subdomains by an incomplete LU factorization. As the crack propagates, we develop a scheme to update the subdomain problems without resolving of them, in which, in order to save the computational cost, only the subdomains around the crack are updated, and all the other regular subdomains remain unchanged. To further speed up the Krylov subspace method, an auxiliary subproblem including crack tips of the previous and current time steps is constructed and solved to provide a better initial guess. We carefully studied the properties of the linear system and the performance of the proposed algorithm. The numerical experiments indicate that the proposed method works well for the simulation of dynamic crack propagation with multiple crack tips.

In this paper, we propose a recycling preconditioning method with auxiliary tip subspace for solving a sequence of highly ill-conditioned linear systems of equations of different sizes arising from elastic crack propagation problems discretized by the extended finite element method. To construct a Schwarz type preconditioner, the finite element mesh is decomposed into crack tip subdomains, which contain all the degrees of freedom (DOFs) of the branch enrichment functions, and regular subdomains, which contain the standard DOFs and the DOFs of the Heaviside and the Junction enrichment functions. As cracks propagate these subdomains are modified accordingly, and the subdomain matrices are constructed as the restriction of the global matrix to the subdomains. In the overlapping Schwarz preconditioners, the crack tip subproblems are solved exactly and the regular subproblems are solved by some inexact solvers, such as ILU. We consider problems with and without crack intersections and develop a simple scheme to update, instead of re-computing, the subdomain problems as cracks propagate, in which only crack tip subdomains are updated around the new crack tips and all the regular subdomains remain unchanged. Therefore, no extra search is required, and the sizes of crack tip subproblems do not increase as cracks propagate, which greatly saves the computational cost. Moreover, starting from the second system, the Krylov subspace method uses a nontrivial initial guess constructed using the solution of the previous system with a modification around the new crack tips. The strategy accelerates the convergence remarkably. Numerical experiments demonstrate the efficiency of the proposed algorithms applied to problems with several types of cracks.

In this paper, we propose some effective one- and two-level domain decomposition preconditioners for elastic crack problems modeled by extended finite element method. To construct the preconditioners, the physical domain is decomposed into the "crack tip" subdomain, which contains all the degrees of freedom (dofs) of the branch enrichment functions, and the "regular" subdomains, which contain the standard dofs and the dofs of the Heaviside enrichment function. In the one-level additive Schwarz and restricted additive Schwarz preconditioners, the "crack tip" subproblem is solved directly and the "regular" subproblems are solved by some inexact solvers, such as ILU. In the two-level domain decomposition preconditioners, traditional interpolations between the coarse and the fine meshes destroy the good convergence property. Therefore, we propose an unconventional approach in which the coarse mesh is exactly the same as the fine mesh along the crack line, and adopt the technique of a non-matching grid interpolation between the fine and the coarse meshes. Numerical experiments demonstrate the effectiveness of the two-level domain decomposition preconditioners applied to elastic crack problems.

In this paper, a conservative parallel iteration scheme is constructed to solve nonlinear diffusion equations on unstructured polygonal meshes. The design is based on two main ingredients: the first is that the parallelized domain decomposition is embedded into the nonlinear iteration; the second is that prediction and correction steps are applied at subdomain interfaces in the parallelized domain decomposition method. A new prediction approach is proposed to obtain an efficient conservative parallel finite volume scheme. The numerical experiments show that our parallel scheme is second-order accurate, unconditionally stable, conservative and has linear parallel speed-up.

In this paper, we are concerned with the constrained finite element method based on domain decomposition satisfying the discrete maximum principle for diffusion problems with discontinuous coefficients on distorted meshes. The basic idea of domain decomposition methods is used to deal with the discontinuous coefficients. To get the information on the interface, we generalize the traditional Neumann-Neumann method to the discontinuous diffusion tensors case. Then, the constrained finite element method is used in each subdomain. Comparing with the method of using the constrained finite element method on the global domain, the numerical experiments show that not only the convergence order is improved, but also the nonlinear iteration time is reduced remarkably in our method.

In this paper, we construct a global repair technique for the finite element scheme of anisotropic diffusion equations to enforce the repaired solutions satisfying the discrete maximum principle. It is an extension of the existing local repair technique. Both of the repair techniques preserve the total energy and are easy to be implemented. The numerical experiments show that these repair techniques do not destroy the accuracy of the finite element scheme, and the computational cost of the global repair technique is cheaper than the local repair technique when the diffusion tensors are highly anisotropic.

In this paper we are concerned with the enhanced strain finite element method for nearly incompressible linear elasticity, in which enhanced strain space plays a key role. We propose a simple and unified way to define the enhanced strain space for k-order finite elements in both two and three dimensions. We show that the approximations generated by the enhanced strain finite element method with the new enhanced strain space possess the optimal error estimates. Moreover, we derive efficient preconditioners for the saddle-point system arising from this enhanced strain finite element method, such that the saddle-point system can be solved in a cheap manner. The numerical results show that our theoretical results are right, and the new method is very effective.

The migration of chemical substances from plastic food packaging into food will endanger consumers' health.The prediction of migration theory is becoming hot spot in recent years.In this paper,the spectral method was used to simulate the mathematical model.The Chebyshev point collocation was given,and the stability and convergence results were analyzed.

In this paper, we propose a domain decomposition method with Lagrange multipliers for three-dimensional linear elasticity, based on geometrically non-conforming subdomain partitions. Some appropriate multiplier spaces are presented to deal with the geometrically non-conforming partitions, resulting in a discrete saddle-point system. An augmented technique is introduced, such that the resulting new saddle-point system can be solved by the existing iterative methods. Two simple inexact preconditioners are constructed for the saddle-point system, one for the displacement variable, and the other for the Schur complement associated with the multiplier variable. It is shown that the global preconditioned system has a nearly optimal condition number, which is independent of the large variations of the material parameters across the local interfaces.

In this paper,we use the domain decomposition method based on geometrically nonconforming decompositions to solve the compressible linear elasticity problems in three dimensions.We prove the numerical solution obtain the optimal error estimate.

(Communicated by Jun Zou) Abstract. In this paper, we propose a domain decomposition method with La- grange multipliers for three-dimensional linear elasticity, based on geometrically non-conforming subdomain partitions. Some appropriate multiplier spaces are presented to deal with the geometrically non-conforming partitions, resulting in a discrete saddle-point system. An augmented technique is introduced, such that the resulting new saddle-point system can be solved by the existing it- erative methods. Two simple inexact preconditioners are constructed for the saddle-point system, one for the displacement variable, and the other for the Schur complement associated with the multiplier variable. It is shown that the global preconditioned system has a nearly optimal condition number, which is independent of the large variations of the material parameters across the local interfaces.