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Full waveform inversion (FWI) entails the ill-posed reconstruction of material parameters (such as sound speed and attenuation) from measurements of complete wave fields (full seismograms). In this paper we present a novel framework for FWI in the visco-acoustic regime. The new framework is based on a new elegant derivation of the system of state and adjoint PDEs which are approximated by the discontinuous Galerkin (DG) method. The inverse problem is then solved by the well established regularization scheme CG-REGINN which has not yet been applied in the context of FWI. For the DG discretization we provide a preconditioner for the efficient computation of the time steps by GMRES which yields optimal convergence estimates in space and time and which is confirmed by numerical tests. The inverse solver expresses the required Fréchet derivative and its adjoint in the DG setting. Successful reconstructions in a simplified cross-well setting serve as a proof of concept for our framework and demonstrate the applicability of our new combination of DG method and inverse solver. Some of the inversion experiments use seismograms generated by an independent finite difference time domain forward solver to avoid inverse crime. [ABSTRACT FROM AUTHOR]

In this paper we consider a family of high-order iterative methods which is more efficient than the Newton method to approximate a solution of symmetric algebraic Riccati equations. In fact, this paper is devoted to the convergence study of a k -steps iterative scheme with low operational cost and high order of convergence. We analyze their accessibility and computational efficiency. We also obtain results about the existence and localization of solution. Numerical experiments confirm the advantageous performance of the iterative scheme analyzed. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the time-domain Maxwell’s equations coupled to another partial differential equation, which arises from modeling of light and structure interaction at the nanoscale. One major contribution of this paper is that the well-posedness is rigorously justified for the first time. Then we propose a fully-discrete finite element method to solve this model. It is interesting to note that we need use curl conforming, divergence conforming, and L 2 finite elements for this model. Numerical stability and optimal error estimate of the scheme are proved. Numerical results justifying our theoretical convergence rate are presented. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the optimal error estimate and the superconvergence of linear fifth order time dependent equations. We prove that the local discontinuous Galerkin (LDG) solution is ( k + 1 ) th order convergent when the piecewise P k space is used. Also, the numerical solution is ( k + 3 2 ) th order superconvergent to a particular projection of the exact solution. The numerical experiences indicate that the order of the superconvergence is ( k + 2 ) , which implies the result obtained in this paper is suboptimal. [ABSTRACT FROM AUTHOR]

Abstract In this paper, by adopting the preconditioned technique for the accelerated generalized successive overrelaxation method (AGSOR) proposed by Edalatpour et al. (2015), we establish the preconditioned AGSOR (PAGSOR) iteration method for solving a class of complex symmetric linear systems. The convergence conditions, optimal iteration parameters and corresponding optimal convergence factor of the PAGSOR iteration method are determined. Besides, we prove that the spectral radius of the PAGSOR iteration method is smaller than that of the AGSOR one under proper restrictions, and its optimal convergence factor is smaller than that of the preconditioned symmetric block triangular splitting (PSBTS) one put forward by Zhang et al. (2018) recently. The spectral properties of the preconditioned PAGSOR matrix are also proposed. Numerical experiments illustrate the correctness of the theories and the effectiveness of the proposed iteration method and the preconditioner for the generalized minimal residual (GMRES) method. [ABSTRACT FROM AUTHOR]

Abstract The construction of finite element approximations in H (div , Ω) usually requires the Piola transformation to map vector polynomials from a master element to vector fields in the elements of a partition of the region Ω. It is known that degradation may occur in convergence order if non affine geometric mappings are used. On this point, we revisit a general procedure for the improvement of two-dimensional flux approximations discussed in a recent paper of this journal (Comput. Math. Appl. 74 (2017) 3283–3295). The starting point is an approximation scheme, which is known to provide L 2 -errors with accuracy of order k + 1 for sufficiently smooth flux functions, and of order r + 1 for flux divergence. An example is R T k spaces on quadrilateral meshes, where r = k or k − 1 if linear or bilinear geometric isomorphisms are applied. Furthermore, the original space is required to be expressed by a factorization in terms of edge and internal shape flux functions. The goal is to define a hierarchy of enriched flux approximations to reach arbitrary higher orders of divergence accuracy r + n + 1 as desired, for any n ≥ 1. The enriched versions are defined by adding higher degree internal shape functions of the original family of spaces at level k + n , while keeping the original border fluxes at level k. The case n = 1 has been discussed in the mentioned publication for two particular examples. General stronger enrichment n > 1 shall be analyzed and applied to Darcy's flow simulations, the global condensed systems to be solved having same dimension and structure of the original scheme. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a nonconforming finite element method (NFEM) is proposed for the constrained optimal control problems (OCPs) governed by a bilinear state equation. The state and adjoint state are approximated by the nonconforming E Q 1 r o t element, and the control is approximated by the orthogonal projection through the state and adjoint state. Some superclose and superconvergence properties are obtained by full use of the distinguish characters of this E Q 1 r o t element, such as the interpolation operator equals the Ritz projection, and the consistency error is one order higher than its interpolation error in the broken energy norm. Finally, some numerical results are provided to verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the robust linearization of nonlinear poromechanics of unsaturated materials. The model of interest couples the Richards equation with linear elasticity equations, generalizing the classical Biot equations. In practice a monolithic solver is not always available, defining the requirement for a linearization scheme to allow the use of separate simulators. It is not met by the classical Newton method. We propose three different linearization schemes incorporating the fixed-stress splitting scheme, coupled with an L-scheme, Modified Picard and Newton linearization of the flow equations. All schemes allow the efficient and robust decoupling of mechanics and flow equations. In particular, the simplest scheme, the Fixed-Stress-L-scheme, employs solely constant diagonal stabilization, has low cost per iteration, and is very robust. Under mild, physical assumptions, it is theoretically shown to be a contraction. Due to possible break-down or slow convergence of all considered splitting schemes, Anderson acceleration is applied as post-processing. Based on a special case, we justify theoretically the general ability of the Anderson acceleration to effectively accelerate convergence and stabilize the underlying scheme, allowing even non-contractive fixed-point iterations to converge. To our knowledge, this is the first theoretical indication of this kind. Theoretical findings are confirmed by numerical results. In particular, Anderson acceleration has been demonstrated to be very effective for the considered Picard-type methods. Finally, the Fixed-Stress-Newton scheme combined with Anderson acceleration shows the best performance among the splitting schemes. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we establish a central limit theorem and a moderate deviation for two-dimensional stochastic primitive equations driven by multiplicative noise. This is the first result about the limit theorem and the moderate deviations for stochastic primitive equations. The proof of the results relies on the weak convergence method and some delicate and careful a p r i o r i estimates. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a new stationary iterative method based on two double splittings of coefficient matrix is proposed to solve linear system A x = b. Some convergence results and comparison theorems of the new method are provided. Theoretical analysis shows that this method is superior to some existing ones under certain conditions. Numerical examples are given to illustrate the theoretical results and examine the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we employ matrix LSQR algorithm to deal with quaternionic least squares problem in order to find the minimum norm solutions with kinds of special structures, and propose a strategy to accelerate convergence rate of the algorithm via right–left preconditioning of the coefficient matrices. We mainly focus on analyzing the minimum norm η -Hermitian solution and the minimum norm η -biHermitian solution to the quaternionic least squares problem, η ∈ { i , j , k }. Other structured solutions also can be obtained using the proposed technique. A number of numerical experiments are performed to show the efficiency of the preconditioned matrix LSQR algorithm. [ABSTRACT FROM AUTHOR]

Abstract In the paper, we present an efficient two grid method for the miscible displacement problem which discretized by mixed finite element methods for the pressure equation and concentration equation at the same time, and then analyzed the error estimate of the two-gird algorithm. At last, the numerical experiment presented confirmed the theoretical results. Compared with the standard mixed finite element methods, this two-grid scheme based on the mixed methods can keep the same convergence order and cost much less work. [ABSTRACT FROM AUTHOR]

Abstract This paper establishes a spectral conjugate gradient method for solving unconstrained optimization problems, where the conjugate parameter and the spectral parameter satisfy a restrictive relationship. The search direction is sufficient descent without restarts in per-iteration. Moreover, this feature is independent of any line searches. Under the standard Wolfe line searches, the global convergence of the proposed method is proved when | β k | ≤ β k F R holds. The preliminary numerical results are presented to show effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract This paper focuses on the unconditional stability and convergence of characteristics type methods for the time-dependent incompressible MHD equations. For this purpose, we introduce a new characteristics time-discrete system. The optimal error estimates in L 2 and H 1 norms for the typical modified characteristics finite element method unconditionally can be deduced, while the whole previous works require certain time-step restrictions. Some numerical experiments document performance of the characteristics type methods for the time-dependent incompressible MHD equations. [ABSTRACT FROM AUTHOR]

Abstract The paper concerns an analysis of an equilibrium problem for 2D elastic body with two semirigid inclusions. It is assumed that inclusions have a joint point, and we investigate a junction problem for these inclusions. The existence of solutions is proved, and different equivalent formulations of the problem are proposed. We investigate a convergence to infinity of a rigidity parameter of the semirigid inclusion. It is proved that in the limit, we obtain an equilibrium problem for the elastic body with a rigid inclusion and a semirigid one. A parameter identification problem is investigated. In particular, the existence of a solution to a suitable optimal control problem is proved. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we present analytical pricing formulae for variance and volatility swaps, when both of the volatility and interest rate are assumed to be stochastic and follow a CIR (Cox–Ingersoll–Ross) process, forming a Heston–CIR hybrid model. The solutions are written in a series form with a theoretical proof of their convergence, ensuring the accuracy of the determined swap prices. The application of the formulae in practice is also demonstrated through the designed numerical experiments. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider the numerical approximation of a general second order semilinear parabolic partial differential equation. Equations of this type arise in many contexts, such as transport in porous media. Using finite element method for space discretization and the exponential Rosenbrock–Euler method for time discretization, we provide a convergence proof in space and time under only the standard Lipschitz condition of the nonlinear part, for both smooth and nonsmooth initial solutions. This is in contrast to restrictive assumptions made in the literature, where the authors have considered only approximation in time so far in their convergence proofs. The main result reveals how the convergence orders in both space and time depend heavily on the regularity of the initial data. In particular, the method achieves optimal convergence order O h 2 + Δ t 2 when the initial data belongs to the domain of the linear operator. Numerical simulations to sustain our theoretical result are provided. [ABSTRACT FROM AUTHOR]

In this paper, a nonlinear control volume finite element (CVFE) scheme for solving an anisotropic degenerate breast cancer development model is introduced and analyzed. This model includes both ordinary differential equations and convection–diffusion–reaction equations modeling the stepwise mutations from a normal breast stem cell to a tumor cell. The diffusion term, which generally involves an anisotropic and heterogeneous diffusion tensor, is discretized on a dual mesh by means of the piecewise linear conforming finite element method and using the Godunov scheme to approximate the diffusion fluxes provided by the conforming finite element reconstruction. The other terms are discretized using a nonclassical upwind finite volume scheme on the dual mesh, where the dual volumes are constructed around the vertices of the original mesh. This technique ensures the positivity and boundedness of discrete solutions without any restriction on the diffusion tensor nor the transmissibility coefficients. The convergence of the scheme is proved, only supposing the shape regularity condition for the original mesh and using a priori estimates as well as the Kolmogorov relative compactness theorem. The proposed scheme is robust, locally conservative, efficient, and stable, which is confirmed by numerical experiments over a general mesh. [ABSTRACT FROM AUTHOR]

An expandable local and parallel two-grid finite element scheme based on superposition principle for elliptic problems is proposed and analyzed in this paper by taking example of Poisson equation. Compared with the usual local and parallel finite element schemes, the scheme proposed in this paper can be easily implemented in a large parallel computer system that has a lot of CPUs. Convergence results based on H 1 and L 2 a priori error estimation of the scheme are obtained, which show that the scheme can reach the optimal convergence orders within | ln H | 2 or | ln H | two-grid iterations if the coarse mesh size H and the fine mesh size h are properly configured in 2-D or 3-D case, respectively. Some numerical results are presented at the end of the paper to support our analysis. [ABSTRACT FROM AUTHOR]

In this paper, by constructing an objective function and using the gradient search, full-rank and reduced-rank gradient-based algorithms are suggested for solving generalized coupled Sylvester matrix equations. It is proved that the reduced-rank iterative algorithm is convergent for proper initial iterative values. By analyzing the spectral radius of the related matrices, the convergence properties are studied and the optimal convergence factor of the reduced-rank algorithm is determined. The relationship between the reduced-rank algorithm and the full-rank algorithm is discussed. Consequently, the computation load can be reduced greatly for solving a class of matrix equation. A numerical example is provided to illustrate the effectiveness of the proposed algorithms and testify the conclusions suggested in this paper. [ABSTRACT FROM AUTHOR]

In this paper, convergence results on the solutions of a time and space discrete model approximation of the Boltzmann equation for a gas of Maxwellian particles in a bounded domain, obtained by Babovsky and Illner (1989), are extended to approximate the solutions of the Boltzmann equation in the whole physical space. This is done for a class of particle interactions including Maxwell and soft cut-off potentials in the sense of Grad. The main result shows that the solutions of the discrete model converge in L 1 to the solutions of the Boltzmann equation, when the discretization parameters go simultaneously to zero. The convergence is uniform with respect to the discretization parameters. In addition, a sufficient condition for the implementation of the main result is provided. The techniques detailed in this paper may be also applied in other contexts. [ABSTRACT FROM AUTHOR]

In this paper a numerical technique is proposed for solving the nonlinear generalized Benjamin-Bona-Mahony-Burgers equation. Firstly, we obtain a time discrete scheme by approximating the first-order time derivative via forward finite difference formula, then we use Kansa's approach to approximate the spatial derivatives. We prove that the time discrete scheme is unconditionally stable and convergent in time variable using the energy method. Also, we show that convergence order of the time discrete scheme is O (τ). We solve the two-dimensional version of this equation using the method presented in this paper on different geometries such as the rectangular, triangular and circular domains and also the three-dimensional case is solved on the cubical and spherical domains. The aim of this paper is to show that the meshless method based on the radial basis functions and collocation approach is also suitable for the treatment of the nonlinear partial differential equations. Also, several test problems including the three-dimensional case are given. Numerical examples confirm the efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, we propose a fully discrete stabilized multiphysics finite element method by the pressure projection method to solve the polymer gel model. To reveal the multi-physical process of the polymer gel model, we reformulate the polymer gel model by introducing a new variable to get a new multi-field problem, then we propose the fully discrete stabilized multiphysics finite element method using the simplest equal order element of P 1 − P 1 − P 1 without the discrete inf–sup condition to solve the new reformulated system, and we prove that the new method not only has no ”locking” phenomenon, but also preserves some energy conservation laws. Also, we prove that the stabilized multiphysics finite element method has optimal convergence order. Finally, we give the numerical example to verify the theoretical results. [ABSTRACT FROM AUTHOR]

Recently, Bai and Benzi proposed a class of regularized Hermitian and skew-Hermitian splitting (RHSS) iteration methods for solving the nonsingular saddle point problem. In this paper, we apply this method to solve the singular saddle point problem. In the process of the semi-convergence analysis, we get that the RHSS method and the HSS method are unconditionally semi-convergent, which has improved the previous results. Then some spectral properties of the corresponding preconditioned matrices and a class of improved preconditioned matrices are analyzed. Finally, some numerical experiments on linear systems arising from the discretization of the Stokes equation are presented to illustrate the feasibility and effectiveness of this method and the corresponding preconditioners. [ABSTRACT FROM AUTHOR]

This paper aims to give an efficient iterative method for solving large sparse nonlinear system with complex symmetric Jacobian matrix. Employing the double-parameter generalized preconditioned MHSS (DGPMHSS) method as the inner iteration, and using the modified Newton method as the outer iteration , we establish a modified Newton–DGPMHSS method for solving nonlinear system with complex symmetric Jacobian matrix. For the new presented method, we provide the local convergence analysis under Hölder condition, which is weaker than Lipschitz condition. Furthermore, we compare our new method with the modified Newton–PMHSS method, which is a considerable method for dealing with large sparse nonlinear system with complex symmetric Jacobian matrix, and the numerical results show the efficiency of our new method. [ABSTRACT FROM AUTHOR]

In this paper, based on the LU decomposition, we propose three non-orthogonal Jacobi-like alternating iterative algorithms with two strategies for solving the joint diagonalization problem of a set of Hermitian matrices. In this kind of algorithm, each transformation includes one upper triangular iterative step and one lower triangular iterative step, and each step involves one parameter. The optimal parameter of each step is derived analytically. The convergence of our proposed algorithms is proven. According to this convergence analysis, the existing GNJD algorithm is revisited. Finally, numerical simulations are presented to illustrate the effectiveness of the proposed algorithms in comparison with existing ones. [ABSTRACT FROM AUTHOR]

Meshfree radial basis function (RBF) methods are popular tools used to numerically solve partial differential equations (PDEs). They take advantage of being flexible with respect to geometry, easy to implement in higher dimensions, and can also provide high order convergence. Since one of the main disadvantages of global RBF-based methods is generally the computational cost associated with the solution of large linear systems, in this paper we focus on a localizing RBF partition of unity method (RBF-PUM) based on a finite difference (FD) scheme. Specifically, we propose a new RBF-PUM-FD collocation method, which can successfully be applied to solve time-dependent PDEs. This approach allows to significantly decrease ill-conditioning of traditional RBF-based methods. Moreover, the RBF-PUM-FD scheme results in a sparse matrix system, reducing the computational effort but maintaining at the same time a high level of accuracy. Numerical experiments show performances of our collocation scheme on two benchmark problems, involving unsteady convection–diffusion and pseudo-parabolic equations. [ABSTRACT FROM AUTHOR]

Nonlocality and spatial heterogeneity of many practical systems have made fractional differential equations very useful tools in Science and Engineering. However, solving these type of models is computationally demanding. In this paper, we propose an exponential integrator method for space fractional models as an attractive and easy-to-code alternative for other existing second-order exponential integrator methods. This scheme is based on using a real distinct poles discretization for the underlying matrix exponentials. One of the major benefits of the proposed scheme is that the algorithm could be easily implemented in parallel to take advantage of multiple processors for increased computational efficiency. The scheme is established to be second-order convergent; and proven to be robust for nonlinear space fractional reaction–diffusion problems involving non-smooth initial data. Our approach is exhibited by solving a system of two-dimensional problems which exhibits pattern formation and has applications in cell-division. Empirically, super-diffusion processes are displayed by investigating the effect of the fractional power of the underlying Laplacian operator on the pattern formation found in these models. Furthermore, the superiority of our method over competing second order ETD schemes, BDF2 scheme, and IMEX schemes is demonstrated. [ABSTRACT FROM AUTHOR]

In this paper, a high-order compact multistep alternating direction implicit (ADI) difference scheme is constructed and analyzed for two-dimensional (2D) nonlinear delayed reaction–diffusion equations (NDREs) with variable coefficients. By the discrete energy method, it is presented that it achieves the convergence rate of O ( τ 2 + h x 4 + h y 4 ) with respect to H 1 - and L 2 -norms in non-constrained temporal grids, and is unconditionally stable. Finally, numerical results demonstrate the effectiveness of our ADI solver and exhibit the correctness of theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we present an improved third-order WENO-Z scheme to improve the order of convergence at critical points. This scheme is constructed through slightly modifying the local smoothness indicators of the conventional WENO-Z scheme with the way of Taylor expansion for third-order convergence. The present scheme is proved to be close to third-order by several standard accuracy tests. The performance enhancement of the WENO scheme through this modification is verified on a variety of one-dimensional and two-dimensional standard numerical experiments. Numerical results indicate that the present scheme provides better results in comparison with the earlier third-order WENO schemes like WENO-JS3, WENO-Z3. [ABSTRACT FROM AUTHOR]

In this paper, an iterative algorithm for solving a generalized coupled Sylvester-conjugate matrix equations over Hermitian R -conjugate matrices given by A 1 V B 1 + C 1 W D 1 = E 1 V ¯ F 1 + G 1 and A 2 V B 2 + C 2 W D 2 = E 2 V ¯ F 2 + G 2 is presented. When these two matrix equations are consistent, the convergence theorem shows that a solution can be obtained within finite iterative steps in the absence of round-off error for any initial arbitrary Hermitian R -conjugate solution matrices V 1 , W 1 . Some lemmas and theorems are stated and proved where the iterative solutions are obtained. A numerical example is given to demonstrate the behavior of the proposed method and to support the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a two-stage method to solve rectangular linear systems that exhibits faster convergence than typical stationary iterative methods. Under suitable conditions, we prove convergence of the new method. The number of outer iterations can be reduced by using a few significant number of inner iterations for efficient computations. Further, we perform a comparison analysis, and establish that a higher number of inner iterations ensures a smaller spectral radius of the global iteration matrix. We also discuss the uniqueness of a proper splitting, and illustrate different comparison theorems for different subclasses of proper splittings. [ABSTRACT FROM AUTHOR]

In this paper, based on a convergence splitting of the matrix A , we present an inner–outer iteration method for solving the linear system A x = b . We analyze the overall convergence of this method without any other restriction on its parameters. Moreover, we give the accelerated inner–outer iteration method, and discuss how to apply the inner–outer iterations as a preconditioner for the Krylov subspace methods. The inner–outer iteration method can also be used for the solution of A X B = C . Finally, several numerical examples are given to validate the performance of our proposed algorithms. [ABSTRACT FROM AUTHOR]

In this paper, a new approach of superconvergent estimate of bilinear finite element is established for two-dimensional time-fractional diffusion equation under fully-discrete scheme. The novelty of this approach is the combination technique of the interpolation and Ritz projection as well as the superclose estimate in H 1 -norm between them, which avoids the difficulty of constructing a postprocessing operator for Ritz projection operator, and reduces the regularity requirement of the exact solution. At the same time, three numerical examples are carried out to verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

Bai (2010) proposed an efficient Hermitian and skew-Hermitian splitting (HSS) iteration method for solving a broad class of large sparse continuous Sylvester equations. To further improve the efficiency of the HSS method, in this paper we present a preconditioned HSS (PHSS) iteration method and its non-alternating variant (NPHSS) for this matrix equation. The convergence properties of the PHSS and NPHSS methods are studied in depth and the quasi-optimal values of the iteration parameters for the two methods are also derived. Moreover, to reduce the computational cost, we establish the inexact variants of the two iteration methods. Numerical experiments illustrate the efficiency and robustness of the two iteration methods and their inexact variants. [ABSTRACT FROM AUTHOR]

In this paper, we consider the numerical methods for stationary Stokes equations with damping. The mark and cell(MAC) method has been applied to discretize the problem on non-uniform grids. We establish the LBB condition and the stability for the MAC scheme. Then we obtain the second order super-convergence in L2 norm for both velocity and pressure on non-uniform grids. We also obtain the second order convergence for some terms of H1 norm of the velocity, and the other terms of H1 norm are second order convergence on uniform grids. Numerical experiments using the MAC scheme show agreement of the numerical results with theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, based on the positive-definite and positive-semidefinite splitting (PPS) iteration scheme, we establish a class of Uzawa-PPS iteration methods for solving nonsingular and singular non-Hermitian saddle point problems with the (1,1) part of the coefficient matrix being non-Hermitian positive definite. Theoretical analyses show that the convergence and semi-convergence properties of the proposed methods can be guaranteed under suitable conditions. Furthermore, we consider acceleration of the Uzawa-PPS methods by Krylov subspace (like GMRES) methods and discuss the spectral properties of the corresponding preconditioned matrix. Numerical experiments are given to confirm the theoretical results which show that the feasibility and effectiveness of the proposed methods and preconditioners. [ABSTRACT FROM AUTHOR]

This paper is presented for the convergence analysis of the element-free Galerkin method for a class of parabolic evolutionary variational inequalities arising from the heat-servo control problem. The error estimates illustrate that the convergence order depends not only on the number of basis functions in the moving least-squares approximation but also the relationship with the time step and the spatial step. Numerical examples verify the convergence analysis and the error estimates. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to analyse two new different possibilities of choosing balanced pairs of approximation spaces for dual (flux) and primal (potential) variables, one for triangles and the other one for quadrilateral elements, to be used in discrete versions of the mixed finite element method for elliptic problems. They can be interpreted as enriched versions of B D F M k + 1 spaces based on triangles, and of R T k spaces for quadrilateral elements. The new flux approximations are incremented with properly chosen internal shape functions (with vanishing normal components on the edges) of degree k + 2 , and matching primal functions of degree k + 1 (higher than the border fluxes, which are kept of degree k ). In all these cases, the divergence of the flux space coincide with the primal approximation space on the master element, producing stable simulations. Using static condensation, the global condensed system to be solved in the enriched cases has same dimension (and structure) of the original ones, which is proportional to the space dimension of the border fluxes for each element geometry. Measuring the errors with L 2 -norms, the enriched space configurations give higher convergence rate of order k + 2 for the primal variable, while keeping the order k + 1 for the flux. For affine meshes, the divergence errors have the same improved accuracy rate as for the error in the primal variable. For quadrilateral non-affine meshes, for instance trapezoidal elements, the divergence error has order k + 1 , one unit more than the order k occurring for R T k spaces on this kind of deformed meshes. This fact also holds for A B F k elements, but for them the potential order of accuracy does not improve, keeping order k + 1 . [ABSTRACT FROM AUTHOR]

By applying the hierarchical identification principle, the gradient-based iterative algorithm is suggested for solving the Sylvester conjugate matrix equation. With the real representation of a complex matrix, a new convergence proof is given. The necessary and sufficient conditions for the convergence factor is determined to guarantee the convergence of the algorithm for any initial iterative matrix. Also a conjecture by Wu et al. (2010) is solved. A numerical example is offered to illustrate the effectiveness of the suggested algorithm and verify some conclusions proposed in this paper. [ABSTRACT FROM AUTHOR]

The consideration of an integral representation as an exact boundary condition for the finite element resolution of wave propagation problems in exterior domain induces algorithmic difficulties. In this paper, we are interested in the resolution of an exterior Maxwell problem in 3D. As a first step, we focus on the justification of an algorithm described in literature, using an interpretation as a Schwarz method. The study of the convergence indicates that it depends significantly on the thickness of the domain of computation. This analysis suggests the use of the finite element term of Schwarz method as a preconditioner for use of Krylov iterative solvers. An analytical study of the case of a spherical perfect conductor indicates the efficiency of such approach. The consideration of the preconditioner suggested by the Schwarz method leads to a superlinear convergence of the GMRES predicted by the analytical study and verified numerically. [ABSTRACT FROM AUTHOR]

This paper shows that the test spaces in discontinuous Petrov Galerkin (DPG) methods can be reduced on rectangular elements without affecting unisolvency or rates of convergences. One reduced case is obtained by decreasing the polynomial degree of a standard test space in both coordinate directions by one. A further reduction of test space by almost another full degree is possible if one is willing to implement a nonstandard test space. The error analysis of such cases is based on an extension of the second Strang lemma and an interpretation of the DPG method as a nonconforming method. The key technical ingredient in obtaining unisolvency is the identification of a discontinuous piecewise polynomial on the element boundary that is orthogonal to all continuous piecewise polynomials of one degree higher. [ABSTRACT FROM AUTHOR]

In this paper, we propose two modified augmented Lagrange multiplier algorithms by mean-value for Toeplitz matrix compressive recovery. In the algorithms, the mean-value modification makes the iteration matrices keep the Toeplitz structure which contributes to reduce the SVD time and CPU time. Numerical experiments show that the proposed algorithms achieve better precision than the augmented Lagrange multiplier method, especially when the matrix E is less sparse. Convergence analysis of the proposed algorithms is also given in detail. [ABSTRACT FROM AUTHOR]

In this paper, an H 1 -Galerkin mixed finite element approximate scheme is established for a class of two-dimensional time fractional diffusion equations by using the bilinear element, Raviart–Thomas element and L 1 time stepping method, which is unconditionally stable and free of LBB condition. And then, without the classical Ritz projection, superclose results for the original variable u in H 1 -norm and the flux p → = ∇ u in H ( d i v , Ω ) -norm are derived by means of properties of the elements and L 1 approximation. Furthermore, with the help of the interpolation postprocessing operator, the global superconvergence results for the original variable u in H 1 -norm are obtained. Finally, numerical simulations verify that the theoretical results are true on both regular meshes and anisotropic meshes. [ABSTRACT FROM AUTHOR]

The adaptive least-squares finite element method (LS-FEM) for the Stokes equations has recently been based on alternative error estimators in Bringmann and Carstensen (2016) for the lowest-order case. Since the first-order div LS-FEM measures the flux errors in H ( div ) , the data resolution error measures the L 2 norm of the right-hand side f minus the piecewise polynomial approximation Π k f without a mesh-size factor. This enforces separate marking with an overall abstract theory (Carstensen and Rabus, 2016). This paper contributes (a) a discussion of the scaling of the LS-FEM, (b) optimal rates of the adaptive h -version of any order k , and (c) comparing numerical results in 2D with all details on inhomogeneous Dirichlet data covered by the analysis. [ABSTRACT FROM AUTHOR]

In this paper, we will develop a new staggered hybridization technique for discontinuous Galerkin methods to discretize linear elastodynamic equations. The idea of hybridization is used extensively in many discontinuous Galerkin methods, but the idea of staggered hybridization is new. Our new approach offers several advantages, namely energy conservation, high-order optimal convergence, preservation of symmetry for the stress tensor, block diagonal mass matrices as well as low dispersion error. The key idea is to use two staggered hybrid variables to enforce the continuity of the velocity and the continuity of the normal component of the stress tensor on a staggered mesh. We prove the stability and the convergence of the proposed scheme in both the semi-discrete and the fully-discrete settings. Numerical results confirm the optimal rate of convergence and show that the method has a superconvergent property for dispersion. Furthermore, an application of this method to Rayleigh wave propagation is presented. [ABSTRACT FROM AUTHOR]

In the present paper we develop the Virtual Element Method for hyperbolic problems on polygonal meshes, considering the linear wave equations as our model problem. After presenting the semi-discrete scheme, we derive the convergence estimates in H 1 semi-norm and L 2 norm. Moreover we develop a theoretical analysis on the stability for the fully discrete problem by comparing the Newmark method and the Bathe method. Finally we show the practical behaviour of the proposed method through a large set of numerical tests. [ABSTRACT FROM AUTHOR]

We consider the solution of block-coupled large-scale linear systems of equations, arising from the finite element approximation of the linear elasticity problem. Due to the large scale of the problems we use properly preconditioned iterative methods, where the preconditioners utilize the underlying block matrix structures, involving inner block solvers and, when suited, broadly established tools such as the algebraic Multigrid method (AMG). For the considered problem, despite of its optimal rate of convergence, AMG, as implemented in some publicly available scientific libraries, imposes unacceptably high demands for computer resources. In this paper we propose and analyze an efficient multilevel preconditioner, based on the Generalized Locally Toeplitz framework, with a specialized transfer operator. We prove and numerically illustrate the optimal convergence rate of the proposed preconditioner, and experimentally report memory and CPU time savings. We also provide comparisons with respect to another aggregation-based algebraic multigrid algorithm. [ABSTRACT FROM AUTHOR]

Classical cascadic multigrid method is optimal under the energy norm for H 2 -regular elliptic problems. This paper analyzes an extrapolation cascadic multigrid (EXCMG) method, originally proposed by Chen et al. (2008) for solving second-order elliptic equations. Following the idea of Bornemann and Deuflhard (1996), we present a superconvergence result for the EXCMG method, which enables us to show that the EXCMG method with the conjugate gradient method as a basic iterative scheme is optimal for H 3 -regular elliptic problems in three dimension with respect to the L 2 -norm. Moreover, we also prove the super-optimality of the EXCMG method under the energy norm for H 2 + α -regular ( 0 < α ≤ 1 ) problems in both two and three dimensions with a reasonable assumption on asymptotic error expansions. Finally, numerical results are presented to verify our theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper illustrates parallel solution of elastoplastic problems with hardening based on the TFETI domain decomposition method with several preconditioning strategies. We consider von Mises plasticity with isotropic hardening using the return mapping concept for one time step. To treat nonlinearity and nonsmoothness, we use the semismooth Newton method. In each Newton iteration, we solve a linear system of equations using the TFETI domain decomposition method with lumped, Dirichlet, or no preconditioner. Our PERMON software is used for the numerical experiments. The observed times and numbers of iterations are backed up by the regular condition number estimates. Both preconditioners accelerate the solution process in terms of the convergence rate. For the worst conditioned benchmark, the most expensive Dirichlet preconditioner gives the lowest computational times. This lets us suggest the Dirichlet preconditioner as the default choice for engineering problems. [ABSTRACT FROM AUTHOR]