A Crank‐Nicolson Petrov‐Galerkin method with quadrature for semi‐linear parabolic problems

We propose and analyze an application of a fully discrete C² spline quadrature Petrov‐Galerkin method for spatial discretization of semi‐linear parabolic initial‐boundary value problems on rectangular domains. We prove second order in time and optimal order H¹ norm convergence in space for the extrapolated Crank‐Nicolson quadrature Petrov‐Galerkin scheme. We demonstrate numerically both L² and H¹ norm optimal order convergence of the scheme even if the nonlinear source term is not smooth. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005.     

Analysis of a two‐level method for anisotropic diffusion equations on aligned and nonaligned grids

This paper is on the convergence analysis for two‐grid and multigrid methods for linear systems arising from conforming linear finite element discretization of the second‐order elliptic equations with anisotropic diffusion. The multigrid algorithm with a line smoother is known to behave well when the discretization grid is aligned with the anisotropic direction; however, this is not the case with a nonaligned grid. The analysis in this paper is mainly focused on two‐level algorithms. For aligned grids, a lower bound is given for a pointwise smoother, and this bound shows a deterioration in the convergence rate, whereas for ‘maximally’ nonaligned grids (with no edges in the triangulation parallel to the direction of the anisotropy), the pointwise smoother results in a robust convergence. With a specially designed block smoother, we show that, for both aligned and nonaligned grids, the convergence is uniform with respect to the anisotropy ratio and the mesh size in the energy norm. The analysis is complemented by numerical experiments that confirm the theoretical results. Copyright © 2012 John Wiley & Sons, Ltd.     

Convergence analysis of the streamline diffusion and discontinuous Galerkin methods for the Vlasov‐Fokker‐Planck system

We prove stability estimates and derive optimal convergence rates for the streamline diffusion and discontinuous Galerkin finite element methods for discretization of the multi‐dimensional Vlasov‐Fokker‐Planck system. The focus is on the theoretical aspects, where we deal with construction and convergence analysis of the discretization schemes. Some related special cases are implemented in M. Asadzadeh [Appl Comput Meth 1(2) (2002), 158–175] and M. Asadzadeh and A. Sopasakis [Comput Meth Appl Mech Eng 191(41–42) (2002), 4641–4661]. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

A cell edge‐based linearity‐preserving scheme for diffusion problems on two‐dimensional unstructured grids

We propose a new finite volume scheme for 2D anisotropic diffusion problems on general unstructured meshes. The main feature lies in the introduction of two auxiliary unknowns on each cell edge, and then the scheme has both cell‐centered primary unknowns and cell edge‐based auxiliary unknowns. The auxiliary unknowns are interpolated by the multipoint flux approximation technique, which reduces the scheme to a completely cell‐centered one. The derivation of the scheme satisfies the linearity‐preserving criterion that requires that a discretization scheme should be exact on linear solutions. The resulting new scheme is then called as a cell edge‐based linearity‐preserving scheme. The optimal convergence rates are numerically obtained on unstructured grids in case that the diffusion tensor is taken to be anisotropic and/or discontinuous. Copyright © 2017 John Wiley & Sons, Ltd.     

A combined BDF‐semismooth Newton approach for time‐dependent Bingham flow

This article is devoted to the numerical simulation of time‐dependent convective Bingham flow in cavities. Motivated by a primal‐dual regularization of the stationary model, a family of regularized time‐dependent problems is introduced. Well posedness of the regularized problems is proved, and convergence of the regularized solutions to a solution of the original multiplier system is verified. For the numerical solution of each regularized multiplier system, a fully discrete approach is studied. A stable finite element approximation in space together with a second‐order backward differentiation formula for the time discretization are proposed. The discretization scheme yields a system of Newton differentiable nonlinear equations in each time step, for which a semismooth Newton algorithm is used. We present two numerical experiments to verify the main properties of the proposed approach. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

A series of high‐order quasi‐compact schemes for space fractional diffusion equations based on the superconvergent approximations for fractional derivatives

Based on the superconvergent approximation at some point (depending on the fractional order α, but not belonging to the mesh points) for Grünwald discretization to fractional derivative, we develop a series of high‐order quasi‐compact schemes for space fractional diffusion equations. Because of the quasi‐compactness of the derived schemes, no points beyond the domain are used for all the high‐order schemes including second‐order, third‐order, fourth‐order, and even higher‐order schemes; moreover, the algebraic equations for all the high‐order schemes have the completely same matrix structure. The stability and convergence analysis for some typical schemes are made; the techniques of treating the fractional derivatives with nonhomogeneous boundaries are introduced; and extensive numerical experiments are performed to confirm the theoretical analysis or verify the convergence orders. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1345–1381, 2015     

A stabilized method for a secondary consolidation Biot's model

This work deals with the numerical solution of a secondary consolidation Biot's model. A family of finite difference methods on staggered grids in both time and spatial variables is considered. These numerical methods use a weighted two‐level discretization in time and the classical central difference discretization in space. A priori estimates and convergence results for displacements and pressure in discrete energy norms are obtained. Numerical examples illustrate the convergence properties of the proposed numerical schemes, showing also a non‐oscillatory behavior of the pressure approximation. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Space‐Time Finite Elements for Fluid‐Structure Interaction

A numerical method for the analysis of fluid‐structure interaction in the areas of building aeroelasticity and building‐water interaction is presented. In order to achieve optimal convergence of the solution a consistent space‐time discretization and a strong coupling algorithm is applied to the highly nonlinear problem.     

Newton‐Krylov iterative matrix representative spectrum

The concept of a representative spectrum is introduced in the context of Newton‐Krylov methods. This concept allows a better understanding of convergence rate accelerating techniques for Krylov‐subspace iterative methods in the context of CFD applications of the Newton‐Krylov approach to iteratively solve sets of non‐linear equations. The dependence of the representative spectrum on several parameters such as mesh, Mach number or discretization techniques is studied and analyzed. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Numerical convergence on adaptive grids for control volume methods

An adaptive technique for control‐volume methods applied to second order elliptic equations in two dimensions is presented. The discretization method applies to initially Cartesian grids aligned with the principal directions of the conductivity tensor. The convergence behavior of this method is investigated numerically. For solutions with low Sobolev regularity, the found L² convergence order is two for the potential and one for the flow density. The system of linear equations is better conditioned for the adaptive grids than for uniform grids. The test runs indicate that a pure flux‐based refinement criterion is preferable.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Eigenvalue approximation of the biharmonic eigenvalue problem by Ciarlet‐Raviart scheme

Using the technique of eigenvalue error expansion and the technique of integral identities, we derive higher order convergence rate of the eigenvalue approximation for the biharmonic eigenvalue problem based on the Ciarlet‐Raviart discretization. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

High‐resolution relaxation scheme for the two‐dimensional Riemann problems in gas dynamics

We present a new relaxation method for the numerical approximation of the two‐dimensional Riemann problems in gas dynamics. The novel feature of the technique proposed here is that it does not require either a Riemann solver or a characteristics decomposition. The high resolution of the method is achieved by using a third‐order reconstruction for the space discretization and a third‐order TVD Runge‐Kutta scheme for the time integration. Numerical experiments, using several configurations of Riemann problems in gas dynamics, are included to confirm the high resolution of the new relaxation scheme. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Fractional stochastic Burgers‐type equation in Hölder space—Wellposedness and approximations

In this work, we use the spectral Galerkin method to prove the existence of a pathwise unique mild solution of a fractional stochastic partial differential equation of Burgers type in a Hölder space. We get the temporal regularity, and using a combination of Galerkin and exponential‐Euler methods, we obtain a full discretization scheme of the solution. Moreover, we calculate the rates of convergence for both approximations (Galerkin and full discretization) with respect to time and to space.     

Analysis of a new BEM‐FEM coupling for two‐dimensional fluid‐solid interaction

In this work we present a new numerical method, based on a coupling of finite and boundary elements, to solve a fluid‐solid interaction problem in the plane. The discrete method uses classical Lagrange finite elements adapted to curved boundaries for the field variable and spectral approximation of the unknowns on the artificial boundary. We provide error estimates for this Galerkin scheme and propose a full discretization based on elementary quadrature formulae, showing that the perturbation due to numerical integration preserves the optimal rate of convergence. We also suggest an iterative method to solve the complicated linear systems arising from this type of schemes. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Stability and convergence of the mark and cell finite difference scheme for Darcy‐Stokes‐Brinkman equations on non‐uniform grids

In this paper, we consider the mark and cell (MAC) method for Darcy‐Stokes‐Brinkman equations and analyze the stability and convergence of the method on nonuniform grids. Firstly, to obtain the stability for both velocity and pressure, we establish the discrete inf‐sup condition. Then we introduce an auxiliary function depending on the velocity and discretizing parameters to analyze the super‐convergence. Finally, we obtain the second‐order convergence in L2 norm for both velocity and pressure for the MAC scheme, when the perturbation parameter ? is not approaching 0. We also obtain the second‐order convergence for some terms of ∥·∥_? norm of the velocity, and the other terms of ∥·∥_? norm are second‐order convergence on uniform grid. Numerical experiments are carried out to verify the theoretical results.     

A linearized Crank–Nicolson–Galerkin FEM for the time‐dependent Ginzburg–Landau equations under the temporal gauge

We propose a decoupled and linearized fully discrete finite element method (FEM) for the time‐dependent Ginzburg–Landau equations under the temporal gauge, where a Crank–Nicolson scheme is used for the time discretization. By carefully designing the time‐discretization scheme, we manage to prove the convergence rate , where τ is the time‐step size and r is the degree of the finite element space. Due to the degeneracy of the problem, the convergence rate in the spatial direction is one order lower than the optimal convergence rate of FEMs for parabolic equations. Numerical tests are provided to support our error analysis. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1279–1290, 2014     

Uncoupling evolutionary groundwater‐surface water flows using the Crank–Nicolson Leapfrog method

Consider an incompressible fluid in a region Ω_f flowing both ways across an interface into a porous media domain Ω_p saturated with the same fluid. The physical processes in each domain have been well studied and are described by the Stokes equations in the fluid region and the Darcy equations in the porous media region. Taking the interfacial conditions into account produces a system with an exactly skew symmetric coupling. Spatial discretization by finite element method and time discretization by Crank–Nicolson LeapFrog give a second‐order partitioned method requiring only one Stokes and one Darcy subphysics and subdomain solver per time step for the fully evolutionary Stokes‐Darcy problem. Analysis of this method leads to a time step condition sufficient for stability and convergence. Numerical tests verify predicted rates of convergence; however, stability tests reveal the problem of growth of numerical noise in unstable modes in some cases. In such instances, the addition of time filters adds stability. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Convergence of multipoint flux approximations on quadrilateral grids

This article presents a convergence analysis of the multipoint flux approximation control volume method, MPFA, in two space dimensions. The MPFA version discussed here is the so‐called O‐method on general quadrilateral grids. The discretization is based on local mappings onto a reference square. The key ingredient in the analysis is an equivalence between the MPFA method and a mixed finite element method, using a specific numerical quadrature, such that the analysis of the MPFA method can be done in a finite element setting. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Semi‐Unstructured Grids in the Laser Simulation Program LASCAD

Automatic grid generation is very important in industrial applications. Furthermore, in several applications discretization grids are needed which provide an anisotropic refinement and an accurate approximation of the derivatives of the solution. These requirements are fulfilled by semi‐unstructured grids. In this paper, we report how semi‐unstructured grids are used in the laser simulation program LASCAD [3].     

Adaptive reduction‐based AMG

With the ubiquity of large‐scale computing resources has come significant attention to practical details of fast algorithms for the numerical solution of partial differential equations. Included in this group are the class of multigrid and algebraic multigrid algorithms that are effective solvers for many of the large matrix problems arising from the discretization of elliptic operators. Algebraic multigrid (AMG) is especially effective for many problems with discontinuous coefficients, discretized on unstructured grids, or over complex geometries. While much effort has been invested in improving the practical performance of AMG, little theoretical understanding of this performance has emerged. This paper presents a two‐level convergence theory for a reduction‐based variant of AMG, called AMGr, which is particularly appropriate for linear systems that have M‐matrix‐like properties. For situations where less is known about the problem matrix, an adaptive version of AMGr that automatically determines the form of the reduction needed by the AMGr process is proposed. The adaptive cycle is shown, in both theory and practice, to yield an effective AMGr algorithm. Copyright © 2006 John Wiley & Sons, Ltd.