Stabilization method for continuous approximation of transient convection problem

The aim of this work is to study a new finite element (FE) formulation for the approximation of nonsteady convection equation. Our approximation scheme is based on the Streamline Upwind Petrov Galerkin (SUPG) method for space variable, x, and a modified of the Euler implicit method for time variable, t. The most interest for this scheme lies in its application to resolve by continuous (FE) method the complex of viscoelastic fluid flow obeying an Oldroyd‐B differential model; this constituted our aim motivation and allows us to treat the constitutive law equation, which expresses the relation between the stress tensor and the velocity gradient and includes tensorial transport term. To make the analysis of the method more clear, we first study, in this article this modified method for the advection equation. We point out the stability of this new method and the error estimate of the approximation solution is discussed. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Stabilized finite element schemes for generalized Oseen systems

In recent works [1, 2], advanced approximation schemes for the numerical calculation of compressible viscous flow were developed, analyzed theoretically and applied successfully to benchmark problems. These methods are based on splitting the Poisson–Stokes system (1.1) describing the motion of a viscous compressible fluid into a generalized Oseen problem for the velocity and a hyperbolic transport equation for the density. Highly refined finite element techniques were proposed for the numerical solution of these separated subproblems of simpler structure; cf. [2]. In this paper, error estimates for a SUPG/(PSPG) and grad-div stabilized finite element approximation of the generalized Oseen problems that arise in the course of the splitting procedure are presented. LBB-stable pairs of finite element spaces are used for velocity and pressure. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fast algorithm for viscous Cahn-Hilliard equation

The main purpose of this paper is to solve the viscous Cahn-Hilliard equation via a fast algorithm based on the two time-mesh (TT-M) finite element (FE) method to ease the problem caused by strong nonlinearities. The TT-M FE algorithm includes the following main computing steps. First, a nonlinear FE method is applied on a coarse time-mesh τ_c. Here, the FE method is used for spatial discretization and the implicit second-order θ scheme (containing both implicit Crank-Nicolson and second-order backward difference) is used for temporal discretization. Second, based on the chosen initial iterative value, a linearized FE system on time fine mesh is solved, where some useful coarse numerical solutions are found by Lagrange’s interpolation formula. The analysis for both stability and a priori error estimates is made in detail. Numerical examples are given to demonstrate the validity of the proposed algorithm. Our algorithm is compared with the traditional Galerkin FE method and it is evident that our fast algorithm can save computational time.     

Local projection FEM stabilization for the time‐dependent incompressible Navier–Stokes problem

We consider conforming finite element (FE) approximations of the time‐dependent, incompressible Navier–Stokes problem with inf‐sup stable approximation of velocity and pressure. In case of high Reynolds numbers, a local projection stabilization method is considered. In particular, the idea of streamline upwinding is combined with stabilization of the divergence‐free constraint. For the arising nonlinear semidiscrete problem, a stability and convergence analysis is given. Our approach improves some results of a recent paper by Matthies and Tobiska (IMA J. Numer. Anal., to appear) for the linearized model and takes partly advantage of the analysis in Burman and Fernández, Numer. Math. 107 (2007), 39–77 for edge‐stabilized FE approximation of the Navier–Stokes problem. Some numerical experiments complement the theoretical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1224–1250, 2015     

Second order fully discrete defect‐correction scheme for nonstationary conduction‐convection problem at high Reynolds number

This survey enfolds rigorous analysis of the defect‐correction finite element (FE) method for the time‐dependent conduction‐convection problem which based on the Crank‐Nicolson scheme. The method consists of two steps: solve a nonlinear problem with an added artificial viscosity term on a FE grid and correct the solutions on the same grid using a linearized defect‐correction technique. The stability and optimal error estimate of the fully discrete scheme are derived. As a consequence, the effectiveness of the method to deal with high Reynolds number is illustrated in several numerical experiments. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 681–703, 2017     

Analysis of linearized Galerkin-mixed FEMs for the time-dependent Ginzburg-Landau equations of superconductivity

A linearized backward Euler Galerkin-mixed finite element method is investigated for the time-dependent Ginzburg-Landau (TDGL) equations under the Lorentz gauge. By introducing the induced magnetic field σ = c u r l A as a new variable, the Galerkin-mixed FE scheme offers many advantages over conventional Lagrange type Galerkin FEMs. An optimal error estimate for the linearized Galerkin-mixed FE scheme is established unconditionally. Analysis is given under more general assumptions for the regularity of the solution of the TDGL equations, which includes the problem in two-dimensional nonconvex polygons and certain three dimensional polyhedrons, while the conventional Galerkin FEMs may not converge to a true solution in these cases. Numerical examples in both two and three dimensional spaces are presented to confirm our theoretical analysis. Numerical results show clearly the efficiency of the mixed method, particularly for problems on nonconvex domains.     

A Two-Grid Finite Element Approximation for Nonlinear Time Fractional Two-Term Mixed Sub-Diffusion and Diffusion Wave Equations

In this paper, we develop a two-grid method (TGM) based on the FEM for 2D nonlinear time fractional two-term mixed sub-diffusion and diffusion wave equations. A two-grid algorithm is proposed for solving the nonlinear system, which consists of two steps: a nonlinear FE system is solved on a coarse grid, then the linearized FE system is solved on the fine grid by Newton iteration based on the coarse solution. The fully discrete numerical approximation is analyzed, where the Galerkin finite element method for the space derivatives and the finite difference scheme for the time Caputo derivative with order $\alpha\in(1,2)$ and $\alpha_{1}\in(0,1)$. Numerical stability and optimal error estimate $O(h^{r+1}+H^{2r+2}+\tau^{\min\{3-\alpha,2-\alpha_{1}\}})$ in $L^{2}$-norm are presented for two-grid scheme, where $t,$ $H$ and $h$ are the time step size, coarse grid mesh size and fine grid mesh size, respectively. Finally, numerical experiments are provided to confirm our theoretical results and effectiveness of the proposed algorithm.     

Convergence of Projected Hessian Approximations in Quasi-Newton Methods for the Nonlinear Programming Problem

Two recently proposed algorithms for the problem of minimizationsubject to nonlinear equality constraints are examined. Bothmaintain quasi-Newton approximations to the projection of theHessian of the Lagrangian, onto the (linearized) manifold ofactive constraints at each iteration. We show that both algorithmscan be used with a more general quasi-Newton updating rule,and, using the analysis of Stoer (1984), that the sequence ofprojected Hessian approximations is convergent.     

Two‐grid methods for semilinear interface problems

In this article, we consider two‐grid finite element methods for solving semilinear interface problems in d space dimensions, for d = 2 or d = 3. We consider semilinear problems with discontinuous diffusion coefficients, which includes problems containing subcritical, critical, and supercritical nonlinearities. We establish basic quasioptimal a priori error estimates for Galerkin approximations. We then design a two‐grid algorithm consisting of a coarse grid solver for the original nonlinear problem, and a fine grid solver for a linearized problem. We analyze the quality of approximations generated by the algorithm and show that the coarse grid may be taken to have much larger elements than the fine grid, and yet one can still obtain approximation quality that is asymptotically as good as solving the original nonlinear problem on the fine mesh. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A fractional step θ-method for convection-diffusion problems

In this article, we analyze the fractional step θ-method for the time-dependent convection-diffusion equation. In our implementation, we completely separate the convection operator from the diffusion operator, and stabilize the convective problem using a Streamline Upwinded Petrov-Galerkin (SUPG) method. We establish a priori error estimates and show that the optimal value of θ yields a scheme that is second-order in time. Numerical computations are presented which demonstrate the method and support the theoretical results.     

A fast interface solver for the biharmonic Dirichlet problem on polygonal domains

Summary. In this paper we propose and analyze an efficient discretization scheme for the boundary reduction of the biharmonic Dirichlet problem on convex polygonal domains. We show that the biharmonic Dirichlet problem can be reduced to the solution of a harmonic Dirichlet problem and of an equation with a Poincaré-Steklov operator acting between subspaces of the trace spaces. We then propose a mixed FE discretization (by linear elements) of this equation which admits efficient preconditioning and matrix compression resulting in the complexity . Here is the number of degrees of freedom on the underlying boundary, is an error reduction factor, or for rectangular or polygonal boundaries, respectively. As a consequence an asymptotically optimal iterative interface solver for boundary reductions of the biharmonic Dirichlet problem on convex polygonal domains is derived. A numerical example confirms the theory. Received September 1, 1995 / Revised version received February 12, 1996     

A virtual element method-based positivity-preserving conservative scheme for convection–diffusion problems on polygonal meshes

In this paper, we propose a positivity-preserving conservative scheme based on the virtual element method (VEM) to solve convection–diffusion problems on general meshes. As an extension of finite element methods to general polygonal elements, the VEM has many advantages such as substantial mathematical foundations, simplicity in implementation. However, it is neither positivity-preserving nor locally conservative. The purpose of this article is to develop a new scheme, which has the same accuracy as the VEM and preserves the positivity of the numerical solution and local conservation on primary grids. The first step is to calculate the cell-vertex values by the lowest-order VEM. Then, the nonlinear two-point flux approximations are utilized to obtain the nonnegativity of cell-centered values and the local conservation property. The new scheme inherits both advantages of the VEM and the nonlinear two-point flux approximations. Numerical results show that the new scheme can reach the optimal convergence order of the virtual element theory, that is, the second-order accuracy for the solution and the first-order accuracy for its gradient. Moreover, the obtained cell-centered values are nonnegative, which demonstrates the positivity-preserving property of our new scheme.     

Block SSOR preconditionings for high-order 3D FE systems. III: Incomplete BSSOR preconditionings based on p-partitionings

This paper considers the construction of high quality IBSSOR preconditionings for p-adaptive hierarchical high-order 3D FE systems. Instead of the commonly-adopted domain decomposition approach to the selection of a block partitioning underlying the corresponding block SSOR preconditioning, we introduce the so-called p-partitionings based on subdividing the degrees of freedom in accordance with the levels of p-refinement. A theoretical analysis of IBSSOR preconditionings is presented. The results of numerical experiments with 3D FE systems arising from the hierarchical approximations of order p, 3 ≤ p ≤ 5, of the 3D equilibrium equations for linear elastic orthotropic materials show that the IBSSOR preconditionings suggested are more efficient than the IBSSOR preconditionings based on domain decomposition partitionings. Moreover, we demonstrate that the new preconditionings are more reliable when solving extremely ill-conditioned 3D FE systems. © 1997 John Wiley & Sons, Ltd.     

Convergence of Galerkin approximations for the Kuramoto‐Tsuzuki equation

Standard Galerkin approximations, using smooth splines to solutions of the Kuramoto‐Tsuzuki equation are analyzed. The existence, uniqueness, and convergence of the fully discrete Crank‐Nicolson scheme are discussed. Furthermore, a second‐order convergent linearized Galerkin approximation are derived. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 21, 2005     

Finite element analysis of a coupling eigenvalue problem on overlapping domains

In this paper, we consider a nonstandard elliptic eigenvalue problem on a rectangular domain, consisting of two overlapping rectangles, where the interaction between the subdomains is expressed through an integral coupling condition on their intersection. For this problem we set up finite element (FE) approximations, without and with numerical quadrature. The involved error analysis is affected by the nonlocal coupling condition, which requires the introduction and error estimation of a suitably modified vector Lagrange interpolant on the overall FE mesh. As a consequence, the resulting error estimates are sub-optimal, as compared to the ones established, e.g., in Vanmaele and van Keer (RAIRO – Math. Mod. Num. Anal 29(3) (1995) 339–365) for classical eigenvalue problems with local boundary or transition conditions.     

Long-Time Accuracy for Approximate Slow Manifolds in a Finite-Dimensional Model of Balance

We study the slow singular limit for planar anharmonic oscillatory motion of a charged particle under the influence of a perpendicular magnetic field when the mass of the particle goes to zero. This model has been used by the authors as a toy model for exploring variational high-order approximations to the slow dynamics in rotating fluids. In this paper, we address the long time validity of the slow limit equations in the simplest nontrivial case. We show that the first-order reduced model remains O(ε) accurate over a long 1/ε timescale. The proof is elementary, but involves subtle estimates on the nonautonomous linearized dynamics.     

Convergence analysis of a projection semi-implicit coupling scheme for fluid–structure interaction problems

In this paper, we provide a convergence analysis of a projection semi-implicit scheme for the simulation of fluid–structure systems involving an incompressible viscous fluid. The error analysis is performed on a fully discretized linear coupled problem: a finite element approximation and a semi-implicit time-stepping strategy are respectively used for space and time discretization. The fluid is described by the Stokes equations, the structure by the classical linear elastodynamics equations (linearized elasticity, plate or shell models) and all changes of geometry are neglected. We derive an error estimate in finite time and we prove that the time discretization error for the coupling scheme is at least ${\sqrt{\delta t}}In this paper, we provide a convergence analysis of a projection semi-implicit scheme for the simulation of fluid–structure systems involving an incompressible viscous fluid. The error analysis is performed on a fully discretized linear coupled problem: a finite element approximation and a semi-implicit time-stepping strategy are respectively used for space and time discretization. The fluid is described by the Stokes equations, the structure by the classical linear elastodynamics equations (linearized elasticity, plate or shell models) and all changes of geometry are neglected. We derive an error estimate in finite time and we prove that the time discretization error for the coupling scheme is at least ?{dt}{\sqrt{\delta t}}. Finally, some numerical experiments that confirm the theoretical analysis are presented.     

Approximation of Singularities by Quantized-Tensor FEM

In d dimensions, first-order tensor-product finite-element (FE) approximations of the solutions of second-order elliptic problems are well known to converge algebraically, with rate at most 1/d in the energy norm and with respect to the number of degrees of freedom. On the other hand, FE methods of higher regularity may achieve exponential convergence, e.g. global spectral methods for analytic solutions and hp methods for solutions from certain countably normed spaces, which may exhibit singularities. In this note, we revisit, in one dimension, the tensor-structured approach to the h-FE approximation of singular functions. We outline a proof of the exponential convergence of such approximations represented in the quantized-tensor-train (QTT) format. Compared to special approximation techniques, such as hp, that approach is fully adaptive in the sense that it finds suitable approximation spaces algorithmically. The convergence is measured with respect to the number of parameters used to represent the solution, which is not the dimension of the first-order FE space, but depends only polylogarithmically on that. We demonstrate the convergence numerically for a simple model problem and find the rate to be approximately the same as for hp approximations. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Normal Multiresolution Approximation of Curves

A multiresolution analysis of a curve is normal if each wavelet detail vector with respect to a certain subdivision scheme lies in the local normal direction. In this paper we study properties such as regularity, convergence, and stability of a normal multiresolution analysis. In particular, we show that these properties critically depend on the underlying subdivision scheme and that, in general, the convergence of normal multiresolution approximations equals the convergence of the underlying subdivision scheme.     

A note on the secant method

It is conjectured that the secant method converges linearly to a root of multiplicitym. In support of this, it is proved that the linearized secant method produces a bounded sequence of approximations except for a restricted set of values of the two initial approximations.