Local Error Estimates Research Articles

Local error estimates and post processing for the Galerkin BEM on polygons

In this paper we give local error estimates in Sobolev norms for the Galerkin method applied to strongly elliptic pseudodifferential equations on a polygon. By using the K-operator, an operator which averages the values of the Galerkin solution, we construct improved approximations.

On the adaptive solution of phase‐field problems with A‐stable explicit last‐stage diagonally implicit Runge–Kutta (ELDIRK) methods

AbstractRunge–Kutta algorithms with adaptive step size control provide reliable tools for the solution of initial value problems with diagonally implicit Runge‐Kutta (DIRK) methods as the most common approach. In this contribution, the new low‐order explicit last‐stage diagonally implicit Runge–Kutta (ELDIRK) methods are investigated, combining implicit schemes with an additional explicit evaluation as an explicit last stage. This results in Butcher tableaus with two solutions of different convergence orders suitable for embedded methods, where the higher‐order solution is achieved by additional explicit evaluations. Thus, the iterative solution of non‐linear systems is omitted for the additional stage, presenting a major reduction in computational cost for the determination of a local error estimate. The key contribution is the application of the novel Butcher tableaux to phase‐field problems, solved with the finite‐element method, leading to substantial numerical investigations with an efficient approach for DIRK schemes. The most important aspects are the investigations of stability properties which lead to the novel class of A‐stable ELDIRK methods. In accordance, the study of the convergence orders is presented. A local error estimator is presented, such that adaptive step size control for the new low‐order embedded schemes based on an empirical approach for error estimation is achieved. A suitable parallel algorithm is presented with conclusive two‐dimensional phase‐field simulations based on a Kobayashi‐Warren‐Carter model. The higher‐order convergence suggested by the novel schemes is confirmed, and their effective results are demonstrated, resulting in a valuable semi‐explicit addition to the family of Runge–Kutta time integration schemes.

Local error estimate of L1 scheme on graded mesh for time fractional Schrödinger equation

Local error estimate of L1 scheme on graded mesh for time fractional Schrödinger equation

Performance Analysis of the Particle Swarm Optimization Algorithm in a VLC System for Localization in Hospital Environments

Localization in hospitals can be valuable in improving different services in medical environments. In this sense, an accurate location system in this environment requires adequately enabling communication technology. However, widely adopted technologies such as Wireless Fidelity (WiFi), Bluetooth, and Radio Frequency Identification (RFID) are considered poorly suited to enable hospital localization due to their inherent drawbacks, including high implementation costs, poor signal strength, imprecise estimates, and potential interference with medical devices. The increasing expenses associated with the implementation and maintenance of these technologies, along with their limited accuracy in dynamic hospital environments, underscore the pressing need for alternative solutions. In this context, it becomes imperative to explore and present novel approaches that not only avoid these challenges but also offer more cost effective, accurate, and interference-resistant connectivity to achieve precise localization within the complex and sensitive hospital environment. In the quest to achieve adequate localization accuracy, this article strategically focuses on leveraging Visible Light Communication (VLC) as a fundamental technology to address the specific demands of hospital environments to achieve the precise localization and tracking of life-saving equipment. The proposed system leverages existing lighting infrastructure and utilizes three transmitting LEDs with different wavelengths. The Received Signal Strength (RSS) is used at the receiver, and a trilateration algorithm is employed to determine the distances between the receiver and each LED to achieve precise localization. The accuracy of the localization is further enhanced by integrating a trilateration algorithm with the sophisticated Particle Swarm Optimization (PSO) algorithm. The proposed method improves the localization accuracy, for example, at a height of 1 m, from a 11.7 cm error without PSO to 0.5 cm with the PSO algorithm. This enhanced accuracy is very important to meet the need for precise equipment location in dynamic and challenging hospital environments to meet the demand for life-saving equipment. Furthermore, the performance of the proposed localization algorithm is compared with conventional positioning methods, which denotes improvements in terms of the localization error and position estimation.

A regional pCO2 climatology of the Baltic Sea from in situ pCO2 observations and a model-based extrapolation approach

Abstract. Ocean surface pCO2 estimates are of great interest for the calculation of air–sea CO2 fluxes, oceanic uptake of anthropogenic CO2, and eventually the Global Carbon Budget. They are accessible from direct observations, which are discrete in space and time and thus always sparse, or from biogeochemical models, which only approximate reality. Here, a combined method for the extrapolation of pCO2 observations is presented that uses (1) model-based patterns of variability from an empirical orthogonal function (EOF) analysis of variability with (2) observational data to constrain EOF pattern amplitudes in (3) an ensemble approach, which locally adjusts the spatial scale of the mapping to the density of the observations. Thus, data-constrained, gap- and discontinuity-free mapped fields including local error estimates are obtained without the need for or dependence on ancillary data (e.g. satellite sea surface temperature maps). This extrapolation approach is generic in that it can be applied to any oceanic or coastal region covered by a suitable model and observations. It is used here to establish a regional pCO2 climatology of the Baltic Sea (Bittig et al., 2023: https://doi.org/10.1594/PANGAEA.961119), largely based on ICOS-DE ship of opportunity (SOOP) Finnmaid surface pCO2 observations between Lübeck-Travemünde (Germany) and Helsinki (Finland). The climatology can serve as improved input for atmosphere–ocean CO2 flux estimation in this coastal environment.

Data assimilation finite element method for the linearized Navier-Stokes equations with higher order polynomial approximation

In this article, we design and analyze an arbitrary-order stabilized finite element method to approximate the unique continuation problem for laminar steady flow described by the linearized incompressible Navier–Stokes equation. We derive quantitative local error estimates for the velocity, which account for noise level and polynomial degree, using the stability of the continuous problem in the form of a conditional stability estimate. Numerical examples illustrate the performances of the method with respect to the polynomial order and perturbations in the data. We observe that the higher order polynomials may be efficient for ill-posed problems, but are also more sensitive for problems with poor stability due to the ill-conditioning of the system.

A numerical method for solving delay and neutral differential equations based on the RKF45 algorithm

For time-dependent or state-dependent delays, the continuously embedded Fehlberg fourth-fifth order Runge-Kutta technique (RKF45) is applied in this study. One advantage of this approach is that it can yield dependable results without requiring consideration of local approximation error and local integration error separately. Additionally, it has been demonstrated that this method can effectively handle derivative discontinuities when solving differential equations with delays. Functional differential equations are a powerful tool for solving problems in fields such as science, engineering, and technology. The delay, neutral differential equations, and functional differential equations in this work are mainly taken from mathematical literature and have been numerically solved using RKF45. The source of the equations and additional pertinent data are also provided. In many cases, graphs have been employed to compare the proposed method with analytical methods and orthogonal neural networks (ONNs) for various applications.

Discretization Error Estimation Using the Unsteady Error Transport Equations

Abstract Computational Fluid Dynamics (CFD) has gained significant utility in the analysis of diverse fluid flow scenarios, thanks to advances in computational power. Accurate error estimation techniques are crucial to ensure the reliability of CFD simulations, as errors can lead to misleading conclusions. This study focuses on the estimation of discretization errors in time-dependent simulations, building upon prior work addressing steady-state problems Wang et al. (2020, “Error Transport Equation Implementation in the Sensei CFD Code,” AIAA Paper No. 2020-1047). In this research, we employ unsteady error transport equations (ETE) to generate localized discretization error estimates within the framework of the finite volume CFD code SENSEI. For steady-state problems, the ETE only need to be solved once after the solution has converged, whereas the unsteady ETE need to be co-advanced with the primal solve. To enhance efficiency, we adopt a one-sided temporal stencil and develop a modified iterative correction process tailored to the unsteady ETE. The time-marching schemes utilized encompass second-order accurate singly diagonally implicit Runge–Kutta (SDIRK) and second-order backward differentiation formula (BDF2), both being implicit. To rigorously assess the accuracy of our error estimates, all test cases feature known analytical solutions, facilitating order-of-accuracy evaluations. Two test cases are considered: the 2D convected vortex for inviscid flow and a cross-term sinusoidal (CTS) manufactured solution for viscous flow. Results indicate higher-order convergence rates for the 2D convected vortex test case even when iterative correction is not applied, with similar observations in the CTS case, albeit not at the finest grid levels. Although the current implementation of iterative correction exhibits lower stability compared to the primal solve, it generally enhances the discretization error estimate. Notably, after iterative correction, the discretization error estimate for the unsteady ETE achieves higher-order accuracy across all grid levels in the 2D CTS manufactured solution.

Implicit integration with a substepping scheme of the zero-elastic range SANISAND-Z model for sand

The SANISAND-Z model is a zero-elastic range boundary surface model. The zero-elastic range makes the model incrementally nonlinear, and consequently, the system of ordinary differential equations of the model formulation is intrinsically implicit. Considering that the calculation results are heavily dependent on the integration step size when the backward Euler method is used to integrate the model, a two-stage diagonally implicit Runge-Kutta algorithm with a substepping scheme is proposed in this study. The substepping scheme is achieved by an embedded local integration error estimate. Based on the user-defined material subroutine interface UMAT of ABAQUS software, the SANISAND-Z model is numerically implemented with the proposed numerical integration algorithm. By monotonic and cyclic element tests and the case of strip footing loading, the performance of the proposed numerical integration algorithm is thoroughly investigated. The results show that the proposed algorithm is more accurate and efficient than the backward Euler method.

On the accuracy, stability and computational efficiency of explicit last-stage diagonally implicit Runge–Kutta methods (ELDIRK) for the adaptive solution of phase-field problems

Runge–Kutta algorithms with adaptive step size control provide reliable tools for the solution of initial value problems. Most widespread within the vast family of Runge–Kutta methods are the diagonally implicit Runge–Kutta (DIRK) methods. In particular, the new low-order explicit last-stage diagonally implicit Runge–Kutta (ELDIRK) methods are investigated, combining implicit schemes with an additional explicit evaluation as an explicit last stage. This results in Butcher tableaus with two solutions of different convergence orders suitable for embedded methods, where the higher-order solution is achieved by additional explicit evaluations. Thus, the iterative solution of non-linear systems is omitted for the additional stage, presenting a major reduction in computational cost for the determination of a local error estimate. The key contribution is the application of the novel Butcher tableaus to phase-field problems, solved with the finite-element method, leading to substantial numerical investigations with an efficient approach for diagonally implicit Runge–Kutta schemes. The most important aspects are the extension towards fourth-order methods, the study of the convergence orders for the ELDIRK schemes, their respective stability regions and computational efficiency. A local error estimator is presented capturing the evolution of the phase-field problems, such that adaptive step size control for the new low-order embedded schemes based on an empirical approach for error estimation is achieved. A suitable parallel algorithm is presented with conclusive two-dimensional phase-field simulations based on a Kobayashi–Warren–Carter model including benchmarks for computational efficiency. The higher-order convergence suggested by the novel schemes is confirmed, and their effective results are demonstrated, resulting in a valuable semi-explicit addition to the family of Runge–Kutta time integration schemes.

Toward a simple and accurate Lagrangian-based error estimator for the BDF algorithms and adaptive time-stepping

PurposeThe purpose of this paper is to design a simple and accurate a-posteriori Lagrangian-based error estimator is developed for the class of backward differentiation formula (BDF) algorithms with variable time step size, and the adaptive time-stepping in BDF algorithms is demonstrated for efficient time-dependent simulations in fluid flow and heat transfer.Design/methodology/approachThe Lagrange interpolation polynomial is used to predict the time derivative, and then the accurate primary result is obtained by the Gauss integral, which is applied to evaluate the local error. Not only the generalized formula of the proposed error estimator is presented but also the specific expression for the widely applied BDF1/2/3 is illustrated. Two essential executable MATLAB functions to implement the proposed error estimator are appended for practical applications. Then, the adaptive time-stepping is demonstrated based on the newly proposed error estimator for BDF algorithms.FindingsThe validation tests show that the newly proposed error estimator is accurate such that the effectivity index is always close to unity for both linear and nonlinear problems, and it avoids under/overestimation of the exact local error. The applications for fluid dynamics and coupled fluid flow and heat transfer problems depict the advantage of adaptive time-stepping based on the proposed error estimator for time-dependent simulations.Originality/valueIn contrast to existing error estimators for BDF algorithms, the present work is more accurate for the local error estimation, and it can be readily extended to practical applications in engineering with a few changes to existing codes, contributing to efficient time-dependent simulations in fluid flow and heat transfer.

Interpolation operators for parabolic problems

We introduce interpolation operators with approximation and stability properties suited for parabolic problems in primal and mixed formulations. We derive localized error estimates for tensor product meshes (occurring in classical time-marching schemes) as well as locally in space-time refined meshes.

Local error estimates of the fourth-order compact difference scheme for a time-fractional diffusion-wave equation

This paper considers the numerical approximation for the time-fractional diffusion-wave equation with the order α∈(1,2), whose solution behaves a weak singularity at t=0. To construct the high-order scheme, the intermediate variable w=Dtα/2(u−tϕ˜) is introduced, then we can rewrite the original problem as a coupled system, where u is the solution of the time-fractional diffusion-wave equation and ϕ˜=ut(x,0). By using the L1 scheme on graded meshes in temporal direction and the compact difference scheme in spatial direction, a fully discrete scheme is constructed for the coupled system. Furthermore, the stability and the local error estimate of the proposed scheme is given. Finally, numerical examples are presented to verify the accuracy and efficiency of the proposed method.

Embedded methods with two-sided local error estimation for solving nonlinear integro-differential equations

Embedded methods with two-sided local error estimation for solving nonlinear integro-differential equations

Isotropic and anisotropic mesh adaptation for RANS simulations of a nacelle under crosswind conditions

Mesh adaptation of unstructured meshes for aerodynamic simulations, that typically resolve the Reynolds Averaged Navier-Stokes (RANS) equations, is a promising approach to enable high numerical precision on complex geometries. Its objective is to minimize the discretization error without using empirical meshing guidelines. The most common approach of mesh adaptation is the “feature-based” isotropic mesh adaptation: from an initial flow prediction on an isotropic unstructured mesh, a local error estimator is computed using a flow variable. It is then used to adapt the mesh using isotropic tetrahedra. Additional near-wall resolution can be achieved by extruding prism layers from the walls. A more efficient approach is to use anisotropic mesh adaptation purely with tetrahedra that are stretched to follow the flow's preferential directions. In this work, we demonstrate the abilities of feature-based isotropic and anisotropic mesh adaptation on a complex flow phenomenon of importance for jet engines: flow separation in a nacelle under crosswind conditions. Two different solvers, adapted for either isotropic or anisotropic meshes, are employed. Results are compared with standard unstructured simulations with user-imposed mesh refinements and highlight the ability of mesh adaptation to automatically capture all the relevant flow phenomena without any user input and at reduced mesh size.

Wavenumber Explicit Convergence of a Multiscale Generalized Finite Element Method for Heterogeneous Helmholtz Problems

In this paper, a generalized finite element (FE) method with optimal local approximation spaces for solving high-frequency heterogeneous Helmholtz problems is systematically studied. The local spaces are built from selected eigenvectors of carefully designed local eigenvalue problems defined on generalized harmonic spaces. At both continuous and discrete levels, (i) wavenumber explicit and nearly exponential decay rates for local and global approximation errors are obtained without any assumption on the size of subdomains and (ii) a quasi-optimal convergence of the method is established by assuming that the size of subdomains is ( is the wavenumber). A novel resonance effect between the wavenumber and the dimension of local spaces on the decay of error with respect to the oversampling size is implied by the analysis. Furthermore, for fixed dimensions of local spaces, the discrete local errors are proved to converge as ( denoting the mesh size) toward the continuous local errors. The method at the continuous level extends the plane wave partition of unity method [I. Babuška and J. M. Melenk, Internat. J. Numer. Methods Engrg., 40 (1997), pp. 727–758] to the heterogeneous-coefficients case, and at the discrete level, it delivers an efficient noniterative domain decomposition method for solving discrete Helmholtz problems resulting from standard FE discretizations. Numerical results are provided to confirm the theoretical analysis and to validate the proposed method.

Localized pointwise error estimates for hybrid finite element methods

We present localized pointwise error estimates for a numerical method designed for accurate and efficient approximations for the flux variables. The method can be considered as a reduced version of the mixed finite element method. The method uses much less degrees of freedom compared to the mixed finite element methods while produces approximation solutions for the flux variable of the same accuracy. In this paper, we present optimal pointwise error estimates for the method, and they show local dependence of the error at a point and weaker dependence on global norms.

Local Error Estimate of an L1-Finite Difference Scheme for the Multiterm Two-Dimensional Time-Fractional Reaction–Diffusion Equation with Robin Boundary Conditions

In this paper, the numerical method for a multiterm time-fractional reaction–diffusion equation with classical Robin boundary conditions is considered. The full discrete scheme is constructed with the L1-finite difference method, which entails using the L1 scheme on graded meshes for the temporal discretisation of each Caputo fractional derivative and using the finite difference method on uniform meshes for spatial discretisation. By dealing with the discretisation of Robin boundary conditions carefully, sharp error analysis at each time level is proven. Additionally, numerical results that can confirm the sharpness of the error estimates are presented.

A variable time-step IMEX-BDF2 SAV scheme and its sharp error estimate for the Navier–Stokes equations

We generalize the implicit-explicit (IMEX) second-order backward difference (BDF2) scalar auxiliary variable (SAV) scheme for Navier–Stokes equation with periodic boundary conditions (Huang and Shen, SIAM J. Numer. Anal. 59 (2021) 2926–2954) to a variable time-step IMEX-BDF2 SAV scheme, and carry out a rigorous stability and convergence analysis. The key ingredients of our analysis are a new modified discrete Grönwall inequality, exploration of the discrete orthogonal convolution (DOC) kernels, and the unconditional stability of the proposed scheme. We derive global and local optimal H1 error estimates in 2D and 3D, respectively. Our analysis provides a theoretical support for solving Navier–Stokes equations using variable time-step IMEX-BDF2 SAV schemes. We also design an adaptive time-stepping strategy, and provide ample numerical examples to confirm the effectiveness and efficiency of our proposed methods.

An Adaptive Time-Stepping Control Algorithm for Molten Salt Reactor Transient Analyses

This paper presents an adaptive time-stepping (ATS) control algorithm developed for transient analyses of Molten Salt Reactors (MSRs) with flowing fuel. To enhance the computational efficiency while maintaining a targeted accuracy, an ATS algorithm with the proportional-integral-derivative (PID) controller and error monitoring was developed and implemented into the PROTEUS-NODAL code. The performance of the ATS algorithm was tested for a wide range of the Molten Salt Fast Reactor (MSFR) numerical benchmark and the Molten Salt Reactor Experiments (MSRE). The test results show over 20 times speedups for very rapid transients and very slow, long-time transients relative to the reference solution obtained with a small time-step size. In addition, extensive parametric studies were performed on the local error estimates and controller parameters and derived a set of recommended values of initial time step, tolerance and gain for four different transient categories.