Maximum-norm Stability Research Articles

An efficient two-grid high-order compact difference scheme with variable-step BDF2 method for the semilinear parabolic equation

Due to the lack of corresponding analysis on appropriate mapping operator between two grids, high-order two-grid difference algorithms are rarely studied. In this paper, we firstly discuss the boundedness of a local bi-cubic Lagrange interpolation operator. And then, taking the semilinear parabolic equation as an example, we first construct a variable-step high-order nonlinear difference algorithm using compact difference technique in space and the second-order backward differentiation formula with variable temporal stepsize in time. With the help of discrete orthogonal convolution kernels, temporal-spatial error splitting idea and a cut-off numerical technique, the unique solvability, maximum-norm stability and corresponding error estimate of the high-order nonlinear difference scheme are established under assumption that the temporal stepsize ratio satisfies rk := τk/τk−1 < 4.8645. Then, an efficient two-grid high-order difference algorithm is developed by combining a small-scale variable-step high-order nonlinear difference algorithm on the coarse grid and a large-scale variable-step high-order linearized difference algorithm on the fine grid, in which the constructed piecewise bi-cubic Lagrange interpolation mapping operator is adopted to project the coarse-grid solution to the fine grid. Under the same temporal stepsize ratio restriction rk < 4.8645 on the variable temporal stepsize, unconditional and optimal fourth-order in space and second-order in time maximum-norm error estimates of the two-grid difference scheme is established. Finally, several numerical experiments are carried out to demonstrate the effectiveness and efficiency of the proposed scheme.

Maximum-norm stability of the finite element method for the Neumann problem in nonconvex polygons with locally refined mesh

The Galerkin finite element solutionuhu_hof the Poisson equation−Δu=f-\Delta u=funder the Neumann boundary condition in a possibly nonconvex polygonΩ\varOmega, with a graded mesh locally refined at the corners of the domain, is shown to satisfy the following maximum-norm stability:‖uh‖L∞(Ω)≤Cℓh‖u‖L∞(Ω),\begin{align*} \|u_h\|_{L^{\infty }(\varOmega )} \le C\ell _h\|u\|_{L^{\infty }(\varOmega )} , \end{align*}whereℓh=ln⁡(2+1/h)\ell _h = \ln (2+1/h)for piecewise linear elements andℓh=1\ell _h=1for higher-order elements. As a result of the maximum-norm stability, the following best approximation result holds:‖u−uh‖L∞(Ω)≤Cℓh‖u−Ihu‖L∞(Ω),\begin{align*} \|u-u_h\|_{L^{\infty }(\varOmega )} \le C\ell _h\|u-I_hu\|_{L^{\infty }(\varOmega )} , \end{align*}whereIhI_hdenotes the Lagrange interpolation operator onto the finite element space. For a locally quasi-uniform triangulation sufficiently refined at the corners, the above best approximation property implies the following optimal-order error bound in the maximum norm:‖u−uh‖L∞(Ω)≤{Cℓhhk+2−2p‖f‖Wk,p(Ω)amp;if r≥k+1,Cℓhhk+1‖f‖Hk(Ω)amp;if r=k,\begin{align*} \|u-u_h\|_{L^\infty (\varOmega )} \le \begin {cases} C\ell _h h^{k+2-\frac {2}{p}} \|f\|_{W^{k,p}(\varOmega )} &\text {if $r\ge k+1$}, \\ C\ell _h h^{k+1} \|f\|_{H^{k}(\varOmega )} &\text {if $r=k$}, \end{cases} \end{align*}wherer≥1r\ge 1is the degree of finite elements,kkis any nonnegative integer no larger thanrr, andp∈[2,∞)p\in [2,\infty )can be arbitrarily large.

Maximal regularity of multistep fully discrete finite element methods for parabolic equations

AbstractThis article extends the semidiscrete maximal $L^p$-regularity results in Li (2019, Analyticity, maximal regularity and maximum-norm stability of semi-discrete finite element solutions of parabolic equations in nonconvex polyhedra. Math. Comp., 88, 1--44) to multistep fully discrete finite element methods for parabolic equations with more general diffusion coefficients in $W^{1,d+\beta }$, where $d$ is the dimension of space and $\beta>0$. The maximal angles of $R$-boundedness are characterized for the analytic semigroup $e^{zA_h}$ and the resolvent operator $z(z-A_h)^{-1}$, respectively, associated to an elliptic finite element operator $A_h$. Maximal $L^p$-regularity, an optimal $\ell ^p(L^q)$ error estimate and an $\ell ^p(W^{1,q})$ estimate are established for fully discrete finite element methods with multistep backward differentiation formulae.

Error estimates for fully discrete BDF finite element approximations of the Allen–Cahn equation

Abstract For a class of compatible profiles of initial data describing bulk phase regions separated by transition zones, we approximate the Cauchy problem of the Allen–Cahn (AC) phase field equation by an initial-boundary value problem in a bounded domain with the Dirichlet boundary condition. The initial-boundary value problem is discretized in time by the backward difference formulae (BDF) of order $1\leqslant q\leqslant 5$ and in space by the Galerkin finite element method of polynomial degree $r-1$, with $r\geqslant 2$. We establish an error estimate of $O(\tau ^q\varepsilon ^{-q-\frac 12}+h^{r}\varepsilon ^{-r-\frac 12}+{e}^{-c/\varepsilon })$ with explicit dependence on the small parameter $\varepsilon$ describing the thickness of the phase transition layer. The analysis utilizes the maximum-norm stability of BDF and finite element methods with respect to the boundary data, the discrete maximal $L^p$-regularity of BDF methods for parabolic equations and the Nevanlinna–Odeh multiplier technique combined with a time-dependent inner product motivated by a spectrum estimate of the linearized AC operator.

Maximum norm error estimates of fourth-order compact difference scheme for the nonlinear Schr\xf6dinger equation involving a quintic term

A compact finite difference (CFD) scheme is presented for the nonlinear Schrödinger equation involving a quintic term. The two discrete conservative laws are obtained. The unconditional stability and convergence in maximum norm with order O({tau }^{2}+h^{4}) are proved by using the energy method. A numerical experiment is presented to support our theoretical results.

A non-polynomial numerical scheme for fourth-order fractional diffusion-wave model

In this paper, we tackle the numerical treatment of a fourth-order fractional diffusion-wave problem. By using parametric quintic spline in the spatial dimension and an approximation of Caputo derivatives at half-points, we propose a numerical scheme and rigorously prove its solvability, convergence and stability in maximum norm. It is shown that the theoretical convergence order improves those of earlier work. To confirm, simulation is carried out to demonstrate the numerical efficiency of the proposed scheme as well as the better performance over other methods.

Analyticity, maximal regularity and maximum-norm stability of semi-discrete finite element solutions of parabolic equations in nonconvex polyhedra

In general polygons and polyhedra, possibly nonconvex, the analyticity of the finite element heat semigroup in the $L^q$ norm, $1\leq q\leq\infty$, and the maximal $L^p$-regularity of semi-discrete finite element solutions of parabolic equations are proved. By using these results, the problem of maximum-norm stability of the finite element parabolic projection is reduced to the maximum-norm stability of the Ritz projection, which currently is known to hold for general polygonal domains and convex polyhedral domains.

Maximum norm stability and error estimates for the evolving surface finite element method

We show convergence in the natural and norm for a semidiscretization with linear finite elements of a linear parabolic partial differential equations on evolving surfaces. To prove this, we show error estimates for a Ritz map, error estimates for the material derivative of a Ritz map and a weak discrete maximum principle.

A high-order finite difference scheme for a singularly perturbed reaction-diffusion problem with an interior layer

In this paper, we consider a singularly perturbed reaction-diffusion problem with a discontinuous source term. Boundary and interior layers appear in the solution. The problem is discretized by using a hybrid finite difference scheme on a Shishkin-type mesh. A nonequidistant generalization of the Numerov scheme is used on the Shishkin-type mesh except for the point of discontinuity, whereas a second-order difference scheme with an additional refined mesh is used for the point of discontinuity. Although the difference scheme does not satisfy the discrete maximum principle, the maximum norm stability of the scheme is established. The maximum error in the mesh points is shown to be uniformly bounded by ( N^{-1}ln N ) ^{4} with a constant independent of the perturbation parameter. Numerical results supporting the theory are presented.

Discrete maximum-norm stability of a linearized second-order finite difference scheme for Allen–Cahn equation

In this paper, we use finite difference methods for solving the Allen–Cahn equation that contains small perturbation parameters and strong nonlinearity. We consider a linearized second-order three-level scheme in time and a second-order finite difference approach in space, and establish discrete boundedness stability in maximum norm: if the initial data are bounded by 1, then the numerical solutions in later times can also be bounded uniformly by 1. It is shown that the main result can be obtained under certain restrictions on the time step.

Numerical approximation of solution for the coupled nonlinear Schrödinger equations

In this article, a compact finite difference scheme for the coupled nonlinear Schrodinger equations is studied. The scheme is proved to conserve the original conservative properties. Unconditional stability and convergence in maximum norm with order O(τ2 + h4) are also proved by the discrete energy method. Finally, numerical results are provided to verify the theoretical analysis.

Discrete maximum-norm stability of a linearized second order finite difference scheme for Allen-Cahn equation

Discrete maximum-norm stability of a linearized second order finite difference scheme for Allen-Cahn equation

Maximum-norm stability of the finite element Ritz projection under mixed boundary conditions

As a model of the second order elliptic equation with non-trivial boundary conditions, we consider the Laplace equation with mixed Dirichlet and Neumann boundary conditions on convex polygonal domains. Our goal is to establish that finite element discrete harmonic functions with mixed Dirichlet and Neumann boundary conditions satisfy a weak (Agmon---Miranda) discrete maximum principle, and then prove the stability of the Ritz projection with mixed boundary conditions in $$L^\infty $$L? norm. Such results have a number of applications, but are not available in the literature. Our proof of the maximum-norm stability of the Ritz projection is based on converting the mixed boundary value problem to a pure Neumann problem, which is of independent interest.

A new numerical scheme for the nonlinear Schrödinger equation with wave operator

In this paper, a new compact finite difference scheme is proposed for a periodic initial value problem of the nonlinear Schrodinger equation with wave operator. This is an explicit scheme of four levels with a discrete conservation law. The unconditional stability and convergence in maximum norm with order $$O(h^{4}+\tau ^{2})$$ are verified by the energy method. Those theoretical results are proved by a numerical experiment and it is also verified that this scheme is better than the previous scheme via comparison.

Maximum-norm stability and maximal $$L^{p}$$ L p regularity of FEMs for parabolic equations with Lipschitz continuous coefficients

In this paper, we study the semi-discrete Galerkin finite element method for parabolic equations with Lipschitz continuous coefficients. We prove the maximum-norm stability of the semigroup generated by the corresponding elliptic finite element operator, and prove the space-time stability of the parabolic projection onto the finite element space in $$L^\infty ( Q_T)$$L?(QT) and $$L^p((0,T);L^q(\Omega ))$$Lp((0,T)?Lq(Ω)), $$1

A compact finite difference scheme for the nonlinear Schrödinger equation with wave operator

In this paper, a compact finite difference scheme is presented for an periodic initial value problem of the nonlinear Schrödinger (NLS) equation with wave operator. This is a scheme of three levels with a discrete conservation law. The unconditional stability and convergence in maximum norm with order O(h4+τ2) are proved by the energy method. A numerical experiment is presented to support our theoretical results.

Robust convergence of a compact fourth-order finite difference scheme for reaction–diffusion problems

We consider a singularly perturbed one-dimensional reaction–diffusion problem with strong layers. The problem is discretized using a compact fourth order finite difference scheme. Altough the discretization is not inverse monotone we are able to establish its maximum-norm stability and to prove its pointwise convergence on a Shishkin mesh. The convergence is uniform with respect to the perturbation parameter. Numerical experiments complement our theoretical results.

Stability and accuracy of adapted finite element methods for singularly perturbed problems

The stability and accuracy of a standard finite element method (FEM) and a new streamline diffusion finite element method (SDFEM) are studied in this paper for a one dimensional singularly perturbed connvection-diffusion problem discretized on arbitrary grids. Both schemes are proven to produce stable and accurate approximations provided that the underlying grid is properly adapted to capture the singularity (often in the form of boundary layers) of the solution. Surprisingly the accuracy of the standard FEM is shown to depend crucially on the uniformity of the grid away from the singularity. In other words, the accuracy of the adapted approximation is very sensitive to the perturbation of grid points in the region where the solution is smooth but, in contrast, it is robust with respect to perturbation of properly adapted grid inside the boundary layer. Motivated by this discovery, a new SDFEM is developed based on a special choice of the stabilization bubble function. The new method is shown to have an optimal maximum norm stability and approximation property in the sense that $$\|u-u_{N}\|_{\infty}\leq C\inf_{v_{N}\in V^{N}}\|u-v_{N}\|_{\infty},$$ where uN is the SDFEM approximation in linear finite element space VN of the exact solution u. Finally several optimal convergence results for the standard FEM and the new SDFEM are obtained and an open question about the optimal choice of the monitor function for the moving grid method is answered.

Maximum-Norm Stability, Smoothing and Resolvent Estimates for Parabolic Finite Element Equations

We survey work on stability and smoothing estimates in maximum-norm for spatially semidiscrete finite element approximations of a model parabolic equation, and related such estimates for the resolvent of the corresponding discrete elliptic operator. We end with a short discussion of stability of fully discrete time stepping methods.

Maximum-norm resolvent estimates for elliptic finite element operators on nonquasiuniform triangulations

In recent years several papers have been devoted to stability and smoothing properties in maximum-norm of finite element discretizations of parabolic problems. Using the theory of analytic semigroups it has been possible to rephrase such properties as bounds for the resolvent of the associated discrete elliptic operator. In all these cases the triangulations of the spatial domain has been assumed to be quasiuniform. In the present paper we show a resolvent estimate, in one and two space dimensions, under weaker conditions on the triangulations than quasiuniformity. In the two-dimensional case, the bound for the resolvent contains a logarithmic factor.