Uniform Error Estimates Research Articles

The TRUNC element in any dimension and application to a modified Poisson equation

AbstractWe introduce a novel TRUNC finite element in dimensions, encompassing the traditional TRUNC triangle as a particular instance. By establishing the weak continuity identity, we identify it as crucial for error estimate. This element is utilized to approximate a modified Poisson equation defined on a convex polytope, originating from the nonlocal electrostatics model. We have substantiated a uniform error estimate and conducted numerical tests on both the smooth solution and the solution with a sharp boundary layer, which align with the theoretical predictions.

Uniform error analysis of an exponential IMEX-SAV method for the incompressible flows with large Reynolds number based on grad-div stabilization

Based on the grad-div stabilization and scalar auxiliary variable (SAV) methods, a first-order Euler implicit/explicit finite element scheme is studied for the Navier–Stokes equations with large Reynolds number. In the designing of numerical scheme, the nonlinear term is explicitly treated such that one only needs to solve a constant coefficient algebraic system at each time step. Meanwhile, the proposed scheme is unconditionally stable without any condition of the time step τ and mesh size h. In finite element discretization, we use the stable Taylor-Hood element for the approximation of the velocity and pressure. By a rigorous analysis, we derive an uniform L2 error estimate O(τ+h2) of the velocity in which the constant is independent of the viscosity coefficient. Finally, numerical experiments are given to support theoretical results and the efficiency of the proposed scheme.

On estimation of the uniform approximation error by interpolating multilinear spline with l p distances

UDC 517.5 We consider the estimation of the uniform approximation error on classes of functions by interpolating multilinear spline with l p distances for p > 3.

Uniform and optimal error estimates of a nested Picard integrator for the nonlinear Schrödinger equation with wave operator

AbstractWe propose a second‐order nested Picard iterative integrator sine pseudospectral (NPI‐SP) method for the nonlinear Schrödinger equation with wave operator (NLSW) involving a parameter and carry out rigorous error estimates. Actually, the equation propagates wave with wavelength in time, while the amplitude of the leading oscillation is for well‐prepared initial data, and for ill‐prepared initial data, respectively. Based on the exponential integrator and nested Picard iteration, the uniformly accurate (w.r.t. ) NPI‐SP scheme is proposed with the optimal uniform error bounds at in time and spectral accuracy in space for both well‐prepared and ill‐prepared data in ‐norm. This result significantly improves the error bounds of the finite difference methods and exponential wave integrator for the NLSW. Error estimates are rigorously carried out and numerical examples are provided to confirm the theoretical analysis.

Uniform weak error estimates for an asymptotic preserving scheme applied to a class of slow-fast parabolic semilinear SPDEs

Uniform weak error estimates for an asymptotic preserving scheme applied to a class of slow-fast parabolic semilinear SPDEs

Robust mixed finite element methods for a quad-curl singular perturbation problem

Robust mixed finite element methods are developed for a quad-curl singular perturbation problem. Lower order H(gradcurl)-nonconforming but H(curl)-conforming finite elements are constructed, which are extended to nonconforming finite element Stokes complexes and the associated commutative diagrams. Then H(gradcurl)-nonconforming finite elements are employed to discretize the quad-curl singular perturbation problem, which possess the sharp and uniform error estimates with respect to the perturbation parameter. The Nitsche’s technique is exploited to achieve the optimal convergence rate in the case of the boundary layers. Numerical results are provided to verify the theoretical convergence rates. In addition, the regularity of the quad-curl singular perturbation problem is established.

An efficient weak Galerkin FEM for third-order singularly perturbed convection-diffusion differential equations on layer-adapted meshes

In this article, we study the weak Galerkin finite element method to solve a class of a third order singularly perturbed convection-diffusion differential equations. Using some knowledge on the exact solution, we prove a robust uniform convergence of order O(N−(k−1/2)) on the layer-adapted meshes including Bakhvalov-Shishkin type, and Bakhvalov-type and almost optimal uniform error estimates of order O((N−1ln⁡N)(k−1/2)) on Shishkin-type mesh with respect to the perturbation parameter in the energy norm using high-order piecewise discontinuous polynomials of degree k. Here N is the number mesh intervals. We conduct numerical examples to support our theoretical results.

The virtual element method with interior penalty for the fourth-order singular perturbation problem

We present the virtual element method with interior penalty to solve a fourth-order singular perturbation problem. In order to estimate the nonconformity error, the degrees of freedom on edges are changed to the moments of functions in the interior penalty scheme. To do this, we design a special H1-type projection that can be uniquely determined by the new degrees of freedom. With the help of the H1-type projection, the local discrete space is defined by imposing some restrictions on the local space of H2-conforming VE, such that it has the local H2 regularity. Then for the numerical method, we derive the error estimates under the condition of enough smooth solution and the uniform error estimates for cases with boundary layers. Finally, we display some numerical examples. From that, we see that the interior penalty method gives a better performance on the convergence than the theoretical predictions.

A numerical technique for solving nonlinear singularly perturbed Fredholm integro-differential equations

This study deals with two numerical schemes for solving a class of singularly perturbed nonlinear Fredholm integro-differential equations. The nonlinear terms are linearized using the quasi-linearization technique. On the layer adapted Shishkin mesh, the numerical solution is initially calculated using the finite difference scheme for the differential part and quadrature rule for the integral part. The method proves to be first-order convergent in the discrete maximum norm. Then, using a post-processing technique we significantly enhance the accuracy from first order to second order. Further, a hybrid scheme on the nonuniform mesh is also constructed and analyzed whose solution converges uniformly, independent of the perturbation parameter and directly gives second order accuracy. Parameter uniform error estimates are demonstrated and the theoretical results are validated through some numerical tests.

A higher order compact numerical approach for singularly perturbed parabolic problem with retarded term

In this work, a compact finite difference approach is constructed for singularly perturbed parabolic reaction diffusion problems with a retarded term. The time and space derivatives have been discretized using the θ-method and a compact fourth-order finite difference method on a Shishkin mesh, respectively. Parameter uniform error estimates have been calculated in the L ∞ norm. Some numerical examples have been considered to corroborate the theoretical results and compare the numerical results with existing methods in the literature. It is shown that the present approach provides improved results till date for the problem considered in this paper.

Improved error estimates of the time‐splitting methods for the long‐time dynamics of the Klein–Gordon–Dirac system with the small coupling constant

AbstractWe provide improved uniform error estimates for the time‐splitting Fourier pseudo‐spectral (TSFP) methods applied to the Klein–Gordon–Dirac system (KGDS) with the small parameter . We first reformulate the KGDS into a coupled Schrödinger–Dirac system (CSDS) and then apply the second‐order Strang splitting method to CSDS with the spatial discretization provided by Fourier pseudo‐spectral method. Based on rigorous analysis, we establish improved uniform error bounds for the second‐order Strang splitting method at up to the long time at . In addition to the conventional analysis methods, we mainly apply the regularity compensation oscillation technique for the analysis of long time dynamic simulation. The numerical results show that our method and conclusion are not only suitable for one‐dimensional problem, but also can be directly extended to higher dimensional problem and highly oscillatory problem. As far as we know there has not been any relevant long time analysis and any improved uniform error bounds for the TSFP method solving the KGDS. Our methods are novel and provides a reference for analyzing the improved error bounds of other coupled systems similar to the KGDS.

Improved uniform error estimates for the two-dimensional nonlinear space fractional Dirac equation with small potentials over long-time dynamics

We develop improved uniform error bounds on a second-order Strang splitting method for the long-time dynamics of the nonlinear space fractional Dirac equation (NSFDE) in two dimension (2D) with small electromagnetic potentials. First, a Strang splitting approach is implemented to discretize NSFDE in time. Afterwards the Fourier pseudospectral method is used to complete the discretization of NSFDE in space. With the aid of a second-order Strang splitting approach employed to the Dirac equation, the major local truncation error of the indicated numerical methods is established. Moreover, for the semi-discrete scheme and full-discretization, we rigorously demonstrate the improved, sharp uniform error estimates are O(ετ2) and O(h1m+h2m+ετ2) in virtue of the regularity compensation oscillation (RCO) technique. In the formulations, τ is the time step, hi(i=1,2) stands for spatial sizes in xi-directions, m is dependent on the regularity of solutions, and ε∈(0,1]. In order to verify our error bounds and to illustrate some fascinating long-time dynamical behaviors of the NSFDE with honeycomb lattice potentials for varied ε, numerical investigations are presented.

A Parameter-Robust Numerical Method Based on Defect-Correction Technique for Parabolic Singular Perturbation Problems with Discontinuous Convection Coefficient and Source

The paper presents a uniformly convergent finite difference method based on the defect correction technique to solve parabolic singular perturbation problems with discontinuous convection coefficient and source. The solution to the problem exhibits interior layer across the discontinuity and demonstrates turning point behavior. The simultaneous presence of perturbation parameters and discontinuity makes the problem stiff. A higher-order method is developed using an implicit difference scheme in time on a uniform mesh and a combination of the upwind difference method and the central difference scheme over an adaptive mesh in space. The method involves iteratively solving increasingly accurate discrete problems by computing and using the defect to correct the approximate solution. Parameter uniform error estimates show uniform convergence of first-order in time and second-order in space. Numerical experiments confirm the accuracy of the proposed scheme and support the theoretical analysis.

A tension spline fitted numerical scheme for singularly perturbed reaction-diffusion problem with negative shift

ObjectiveThe paper is focused on developing and analyzing a uniformly convergent numerical scheme for a singularly perturbed reaction-diffusion problem with a negative shift. The solution of such problem exhibits strong boundary layers at the two ends of the domain due to the influence of the perturbation parameter, and the term with negative shift causes interior layer. The rapidly changing behavior of the solution in the layers brings significant difficulties in solving the problem analytically. We have treated the problem by proposing a numerical scheme using the implicit Euler method in the temporal direction and a fitted tension spline method in the spatial direction with uniform meshes.ResultStability and uniform error estimates are investigated for the developed numerical scheme. The theoretical finding is demonstrated by numerical examples. It is obtained that the developed numerical scheme is uniformly convergent of order one in time and order two in space.

Uniform Error Estimates of the Finite Element Method for the Navier-Stokes Equations in R2 with L2 Initial Data.

In this paper, we study the finite element method of the Navier-Stokes equations with the initial data belonging to the L2 space for all time t>0. Due to the poor smoothness of the initial data, the solution of the problem is singular, although in the H1-norm, when t∈[0,1). Under the uniqueness condition, by applying the integral technique and the estimates in the negative norm, we deduce the uniform-in-time optimal error bounds for the velocity in H1-norm and the pressure in L2-norm.

Parameter uniform fitted mesh finite difference scheme for elliptical singularly perturbed problems with mixed shifts in two dimensions

In this work, we embarked on the research of linear two-dimensional elliptic singularly perturbed problems having positive and negative shifts in both directions of the reaction terms, whose solution exhibits characteristic boundary layers. The study of this class of problems was started in the mid-eighties, but all the studies were restricted to 1D. The Taylor series expansion is used to estimate decelerated terms of the problem and the emerging problem is discretized using the fitted mesh finite difference method to establish parameter uniform error estimates. The effect of positive and negative shifts on the behaviour of the solution is explained by executing numerical experiments on two test examples.

Error estimations of a weak Galerkin finite element method for a linear system of $ \ell\geq 2 $ coupled singularly perturbed reaction-diffusion equations in the energy and balanced norms

<abstract><p>This paper introduces a weak Galerkin finite element method for a system of $ \ell\geq 2 $ coupled singularly perturbed reaction-diffusion problems. The proposed method is independent of parameter and uses piecewise discontinuous polynomials on interior of each element and constant on the boundary of each element. By the Schur complement technique, the interior unknowns can be locally efficiently eliminated from the resulting linear system, and the degrees of freedom of the proposed method are comparable with the classical FEM. It has been reported that the energy norm is not adequate for singularly perturbed reaction-diffusion problems since it can not efficiently reflect the behaviour of the boundary layer parts when the diffusion coefficient is very small. For the first time, the error estimates in the balanced norm has been presented for a system of coupled singularly perturbed problems when each equation has different parameter. Optimal and uniform error estimates have been established in the energy and balanced norm on an uniform Shishkin mesh. Finally, we carry out various numerical experiments to verify the theoretical findings.</p></abstract>

NIPG Method on Shishkin Mesh for Singularly Perturbed Convection–Diffusion Problem with Discontinuous Source Term

In this paper, we provide numerical approaches for singularly perturbed convection–diffusion problem with a discontinuous source term. As a result of the discontinuity in the source term, the problem highlights an exponential-type boundary layer and a weak interior layer. To handle both the layers at the same time, we design a nonsymmetric discontinuous Galerkin finite element technique with interior penalties (NIPG). The problem is solved using the standard Shishkin mesh. In the DG norm, uniform error estimates are derived for the problem. To verify the theoretical estimates, numerical tests are carried out.

A higher‐order hybrid spline difference method on adaptive mesh for solving singularly perturbed parabolic reaction–diffusion problems with robin‐boundary conditions

AbstractWe propose a hybrid numerical scheme to discretize a class of singularly perturbed parabolic reaction–diffusion problems with robin‐boundary conditions on an equidistributed grid. The hybrid difference scheme is developed by using a modified backward difference scheme in time, a combination of the cubic spline and exponential spline difference scheme in space. The proposed scheme uses a cubic spline difference scheme for the discretization of robin‐boundary conditions. For the time discretization of the problem, we use the standard uniform mesh while a layer adapted equidistributed grid is generated for the spatial discretization. By equidistributing a curvature‐based monitor function, the spatial adaptive grid is able to capture the presence of parabolic boundary layers without using any prior information about the solution. Parameter uniform error estimates are derived to illustrate an optimal convergence of first‐order in time and second‐order in space for the proposed discretization. The accuracy of the proposed scheme is confirmed by the numerical experiments that underpin the theoretical analysis.

Uniform Finite Element Error Estimates with Power-Type Asymptotic Constants for Unsteady Navier-Stokes Equations.

Uniform error estimates with power-type asymptotic constants of the finite element method for the unsteady Navier–Stokes equations are deduced in this paper. By introducing an iterative scheme and studying its convergence, we firstly derive that the solution of the Navier–Stokes equations is bounded by power-type constants, where we avoid applying the Gronwall lemma, which generates exponential-type factors. Then, the technique is extended to the error estimate of the long-time finite element approximation. The analyses show that, under some assumptions on the given data, the asymptotic constants in the finite element error estimates for the unsteady Navier–Stokes equations are uniformly power functions with respect to the initial data, the viscosity, and the body force for all time . Finally, some numerical examples are shown to verify the theoretical predictions.