Global Error Estimates Research Articles

A posteriori error estimation and adaptivity based on VMS for the Stokes problem

SummaryIn this paper, we present residual‐based error estimators applied to the Stokes problem. Implicit and explicit error estimators are developed to predict the error of the numerical solution obtained by a stabilized finite element formulation. Both error estimators arise from the variational multiscale framework, in which the variational form is split into coarse scales (finite element method solution) and fine scales (committed error). This leads to a local problem set on each element in which the error and the residuals are involved. The different ways of linking the error to the residuals produce two error estimators. Local and global error estimates measured in the H1‐seminorm are provided for triangles and quadrilaterals. As an application, using the local error estimates, an adaptive mesh refinement strategy is implemented in order to optimize the computational resources.

A posteriori error estimation for incompressible viscous fluid with a new boundary condition

This paper describes numerical solutions of incompressible Navier-Stokes equations with a new boundary condition. To solve this problem, we use the discretization by mixed finite element method. We use a vector extrapolation method for computing numerical solutions of the steady-state Navier-Stokes equations. In addition, two types of a posteriori error indicator are introduced and are shown to give global error estimates that are equivalent to the true error. A numerical experiment on the driven cavity flow is given to demonstrate the effectiveness of the vector extrapolation method. We compare the result with the solution from commercial code like ADINA system as well as with values from other simulations.

Adaptive Interpolation Algorithm Based on a kd-Tree for Numerical Integration of Systems of Ordinary Differential Equations with Interval Initial Conditions

We consider issues related to the numerical solution of interval systems of ordinary differential equations. We suggest an algorithm that permits finding interval estimates of solutions with prescribed accuracy in reasonable time. The algorithm constructs an adaptive partition (a dynamic structured grid) based on a kd-tree over the space formed by interval initial conditions for the ordinary differential equations. In the operation of the algorithm, a piecewise polynomial function interpolating the dependence of the solution on the specific values of interval parameters is constructed at each step of solution of the original problem. We prove that the global error estimate linearly depends on the height of the kd-tree. The algorithm is tested on several examples; the test results show its efficiency when solving problems of the class under study.

Stability and Convergence Analysis of a Class of Continuous Piecewise Polynomial Approximations for Time-Fractional Differential Equations

We propose and study a class of numerical schemes to approximate time-fractional differential equations. The methods are based on the approximations of the Caputo fractional derivative of order alpha in (0, 1) by using continuous piecewise polynomials, which are strongly related to the backward differentiation formulae. We investigate their theoretical properties, such as the local truncation error and global error estimates with respect to sufficiently smooth solutions, and the numerical stability in terms of stability region and A(frac{pi }{2})-stability. Numerical experiments are given to verify our theoretical investigations.

On the Numerical Approximability of Stable Dynamical Systems

This paper discusses the numerical approximation of stable dynamical systems of ordinary differential equations by general time-stepping methods. The traditional error analysis for this approximation yields estimates for the global discretization error in terms of the local truncations errors with constants generally growing exponentially in time due to the use of a discrete Gronwall inequality. In special cases of monotone systems, global error estimates can be shown to hold uniformly in time provided that also the time stepping scheme possesses certain monotonicity properties. The standard example is the backward Euler method. However, for more general time-stepping schemes, this is not clear even in the monotone case. It is shown here how in general situations certain qualitative stability properties (exponential and quasi-exponential stability) of the solution to be approximated can be used for extending local error estimates to global ones holding uniformly in time. Further, the approximate solutions are likewise stable and, in the autonomous case, tend to equilibrium points.

Doubly quasi-consistent fixed-stepsize numerical integration of stiff ordinary differential equations with implicit two-step peer methods

The problem of accurate and efficient numerical integration of ordinary differential equations (ODEs) is a key issue of computation science, which has a valued practical impact in various topics of research. In this respect, effective global error estimation and control strategies are of particular theoretical and practical significance. They allow many applied dynamic systems to be treated numerically for user-supplied accuracy conditions in automatic mode. The property of double quasi-consistency plays an outstanding role in the mentioned topic because the global (or true) error of a doubly quasi-consistent method is asymptotically equal to its local error. Therefore, this entails quite a nontrivial implication that the true error of numerical integration can be cheaply evaluated and regulated within a single solution run. The latter was considered to be impossible for decades. The property of double quasi-consistency goes back to 2009, when Kulikov studied it in the first time but failed to find doubly quasi-consistent numerical methods among Nordsieck formulas. Later on, Kulikov and Weiner proved the existence of doubly quasi-consistent numerical schemes among superconvergent explicit two-step parallel peer methods and examined their practical efficiency in comparison to explicit Matlab ODE solvers in 2010. The focus of the present research is on accurate numerical integration formulas for treating stiff ODEs, which often arise in practice and for which explicit methods are shown to be ineffective. Here, we make the first step towards an accurate and efficient numerical solution of stiff ODEs and prove existence of implicit doubly quasi-consistent formulas. We fulfill our investigation of double quasi-consistency within the family of fixed-stepsize implicit two-step peer schemes and construct two methods of convergence orders 3 and 4, which possess excellent stability properties. Our theoretical results are confirmed with numerical integration of both nonstiff and stiff test problems in this paper.

Adaptive finite element modelling of three-dimensional magnetotelluric fields in general anisotropic media

We present a goal-oriented adaptive finite element (FE) modelling algorithm for 3-D magnetotelluric fields in generally anisotropic conductivity media. The model consists of a background layered structure, containing anisotropic blocks. Each block and layer might be anisotropic by assigning to them 3×3 conductivity tensors. The second-order partial differential equations are solved using the adaptive finite element method (FEM). The computational domain is subdivided into unstructured tetrahedral elements, which allow for complex geometries including bathymetry and dipping interfaces. The grid refinement process is guided by a global posteriori error estimator and is performed iteratively.The system of linear FE equations for electric field E is solved with a direct solver MUMPS. Then the magnetic field H can be found, in which the required derivatives are computed numerically using cubic spline interpolation. The 3-D FE algorithm has been validated by comparisons with both the 3-D finite-difference solution and 2-D FE results. Two model types are used to demonstrate the effects of anisotropy upon 3-D magnetotelluric responses: horizontal and dipping anisotropy. Finally, a 3D sea hill model is modelled to study the effect of oblique interfaces and the dipping anisotropy.

ELEMENTWISE DECOMPOSITION OF A POSTERIORI ERROR ESTIMATOR BASED ON REFERENCE SOLUTION FOR HP-ADAPTIVE FINITE ELEMENT METHOD

We consider the construction of a posteriori error estimator (AEE), based on the reference solution for one-dimensional hp-adaptive finite element method (FEM). It is shown that the square of the global error estimator can be represented as a sum of squares of single element error indicators, which are calculated independently on the elements of the selected mesh. The obtained decomposition can be used to justify the iterative algorithms of hp-adaptive FEM schemes. The proposed estimator can be used to calculate the error decrease rates for different local mesh refinement of just single finite element without re-calculating the entire reference solution on the whole problem domain. In addition, by choosing the system of Lobatto functions as a finite element basis, we propose an efficient scheme for calculating the local indicators. Key words: finite element method, Galerkin method, orthogonal projection, Lobatto basis, Green’s function, hp-adaptivity, a posteriori error estimator, well posed problem, diffusion-advection-reaction boundary value problem, reference solution.

Error analysis of the high order scheme for homogenization of Hamilton–Jacobi equation

In this paper, employing ideas developed for conservation law equations such as the Lax–Friedrich-type and Godunov-type numerical fluxes, we describe the numerical schemes for approximating the solution of the limit problem arising in the homogenization of Hamilton–Jacobi equations. All approximation methods involve three steps. The first scheme is a provably monotonic discretization of the cell problem for approximating the effective Hamiltonian for a given vector P∈RN. Next, using interpolation, we present an approximation of the effective Hamiltonian in the domain RN. Finally, the numerical schemes of the Hamilton–Jacobi equations with the effective Hamiltonian approximation are constructed. We also present global error estimates including all the discrete mesh sizes. The theoretical results are illustrated through numerical examples, including two convex Hamiltonians and two non-convex Hamiltonians.

Numerical treatment and Global Error Estimation of a MHD Flows of an Oldroyd 6-Constant Nano-Fluid through a non-Darcy Porous medium with Heat and Mass Transfer

Explicit Finite-Difference method was used to obtain the solution of the system of the non-linear ordinary differential equations which transform from the non-linear partial differential equations. These equations describe the steady magneto-hydrodynamic flow of an oldroyd 6-constant non-Newtonian nano-fluid through a non-Darcy porous medium with heat and mass transfer. The numerical formula of the velocity, the temperature, the concentration, and the nanoparticles concentration distributions of the problem were illustrated graphically. The effect of Darcy number Da, Forchheimer number Fs, magnetic field parameter M, local temperature Grashof number Gr, local nanoparticle Grashof Br, Prandtl number Pr, Dufour number Nd, Brownian motion parameter Nb, Thermophoresis parameter Nt, Lewis number Le, Sort number Ld, Chemical reaction parameter Rc, and Chemical reaction order m on those formula were discussed at the values of material parameters ( specially in the case of pure Coutte flow. Then, the effects of modified pressure gradients on those formulas were discussed in the case of pure Poiseuille flow and the generalized Couette flow. Also, an estimation of the global error for the numerical values of the solutions is calculated by using Zadunaisky technique.

Generalizing global error estimation for ordinary differential equations by using coupled time-stepping methods

This study introduces new time-stepping strategies with built-in global error estimators. The new methods propagate the defect along with the numerical solution much like solving for the correction or Zadunaisky’s procedure; however, the proposed approach allows for overlapped internal computations and, therefore, represents a generalization of the classical numerical schemes for solving differential equations with global error estimation. The resulting algorithms can be effectively represented as general linear methods. Several explicit self-starting schemes akin to Runge–Kutta methods with global error estimation are introduced, and the theoretical considerations are illustrated in several examples.

Error estimates on a finite volume method for diffusion problems with interface on rectangular grids

The finite volume methods are frequently employed in the discretization of diffusion problems with interface. In this paper, we firstly present a vertex-centered MACH-like finite volume method for solving stationary diffusion problems with strong discontinuity and multiple material cells on the Eulerian quadrilateral grids. This method is motivated by Frese [No. AMRC-R-874, Mission Research Corp., Albuquerque, NM, 1987]. Then, the local truncation error and global error estimates of the degenerate five-point MACH-like scheme are derived by introducing some new techniques. Especially under some assumptions, we prove that this scheme can reach the asymptotic optimal error estimate O(h2|ln h|) in the maximum norm. Finally, numerical experiments verify theoretical results.

A higher order weak approximation scheme of multidimensional stochastic differential equations using Malliavin weights

We show a new higher order weak approximation with Malliavin weights for multidimensional stochastic differential equations by extending the method in Takahashi and Yamada (2016). The estimate of global error of the discretization is based on a sharp small time expansion using a Malliavin calculus approach. We give explicit Malliavin weights for second order discretization as polynomials of Brownian motions. The effectiveness is illustrated through an example in option pricing.

An Improved XFEM for the Poisson Equation with Discontinuous Coefficients

Abstract Discontinuous coefficients in the Poisson equation lead to the weak discontinuity in the solution, e.g. the gradient in the field quantity exhibits a rapid change across an interface. In the real world, discontinuities are frequently found (cracks, material interfaces, voids, phase-change phenomena) and their mathematical model can be represented by Poisson type equation. In this study, the extended finite element method (XFEM) is used to solve the formulated discontinuous problem. The XFEM solution introduce the discontinuity through nodal enrichment function, and controls it by additional degrees of freedom. This allows one to make the finite element mesh independent of discontinuity location. The quality of the solution depends mainly on the assumed enrichment basis functions. In the paper, a new set of enrichments are proposed in the solution of the Poisson equation with discontinuous coefficients. The global and local error estimates are used in order to assess the quality of the solution. The stability of the solution is investigated using the condition number of the stiffness matrix. The solutions obtained with standard and new enrichment functions are compared and discussed.

Fully decoupled monolithic projection method for natural convection problems

To solve time-dependent natural convection problems, we propose a fully decoupled monolithic projection method. The proposed method applies the Crank–Nicolson scheme in time and the second-order central finite difference in space. To obtain a non-iterative monolithic method from the fully discretized nonlinear system, we first adopt linearizations of the nonlinear convection terms and the general buoyancy term with incurring second-order errors in time. Approximate block lower-upper decompositions, along with an approximate factorization technique, are additionally employed to a global linearly coupled system, which leads to several decoupled subsystems, i.e., a fully decoupled monolithic procedure. We establish global error estimates to verify the second-order temporal accuracy of the proposed method for velocity, pressure, and temperature in terms of a discrete l2-norm. Moreover, according to the energy evolution, the proposed method is proved to be stable if the time step is less than or equal to a constant. In addition, we provide numerical simulations of two-dimensional Rayleigh–Bénard convection and periodic forced flow. The results demonstrate that the proposed method significantly mitigates the time step limitation, reduces the computational cost because only one Poisson equation is required to be solved, and preserves the second-order temporal accuracy for velocity, pressure, and temperature. Finally, the proposed method reasonably predicts a three-dimensional Rayleigh–Bénard convection for different Rayleigh numbers.

Efficient goal-oriented global error estimators for BDF methods using discrete adjoints

In this paper we develop new goal-oriented global error estimators for variable multistep backward differentiation formulae (BDF) methods. These estimators use, for the first time in the context of multistep methods, discrete adjoints computed by adjoint differentiation of the nominal integration scheme. The derivation is based on the recently developed Petrov–Galerkin finite element formulation of BDF methods and their discrete adjoint schemes and makes use of the dual weighted residual methodology. Defect integrals or local truncation errors are used as local error quantities. We prove the asymptotic correctness and optimal convergence of the novel estimators for the one-step BDF method. For multistep BDF methods using a selfstarting procedure we show that the estimator with defect integrals converges but suboptimal and that its effectivity index converges to an offset value. On the other hand, the estimator with local truncation errors is again asymptotically correct and of optimal order. We confirm these results numerically. With a real-world example from chemical engineering, we give promising numerical results for the estimation accuracy in variable BDF-type methods with changing orders and stepsizes. Finally, we give a first use of the novel estimators for goal-oriented global error control of a challenging stiff test problem.

Analysis of velocity-components decoupled projection method for the incompressible Navier–Stokes equations

We study the temporal accuracy and stability of the velocity-components decoupled projection method (VDPM) for fully discrete incompressible Navier–Stokes equations. In particular, we investigate the effect of three formulations of the nonlinear convection term, which include the advective, skew-symmetric, and divergence forms, on the temporal accuracy and stability. Second-order temporal accuracy of the VDPM for both velocity and pressure is verified by establishing global error estimates in terms of a discrete l2-norm. Considering the energy evolution, we demonstrate that the VDPM is stable when the time step is less than or equal to a constant. Stability diagrams, which display the distributions of the maximum magnitude of the eigenvalues of the corresponding amplification matrices, are obtained using von Neumann analysis. These diagrams indicate that the advective form is more stable than the other formulations of the nonlinear convection term. Numerical tests are performed in order to support the mathematical findings involving temporal accuracy and stability, and the effects of the formulations of the nonlinear convection term are analyzed. Overall, our results indicate that the VDPM along with an advective discrete convection operator is almost unconditionally stable, second-order accurate in time, and computationally efficient because of the non-iterative solution procedure in solving the decoupled momentum equations.

A posteriori global error estimator based on the error in the constitutive relation for reduced basis approximation of parametrized linear elastic problems

In this paper we introduce a posteriori error estimator based on the concept of error in the constitutive relation to verify parametric models computed with a reduced basis approximation. We develop a global error estimator which leads to an upper bound for the exact error and takes into account all the error sources: the error due to the reduced basis approximation as well as the error due to the finite element approximation. We propose an error indicator to measure the quality of the reduced basis approximation and we deduce an error indicator on the finite element approximation.

Interpolation and Sampling Errors of the Ash and Sulphur Contents in Selected Polish Bituminous Coal Deposit (Upper Silesian Coal Basin – USCB) / Błędy Interpolacji I Opróbowania Zawartości Popiołu I Siarki W Wytypowanych Polskich Złożach Węgla Kamiennego (Górnośląskie Zagłębie Węglowe)

Abstract The basic sources of information on the parameters characterizing the quality of coal (i.e. its ash and sulphur contents) in the deposits of The Upper Silesian Coal Basin (Poland) are drill core sampling (the first stage of exploration) and channel sampling in mine workings (the second stage of exploration). Boreholes are irregularly spaced but provide relatively uniform coverage over an entire deposit area. Channel samples are taken regularly in mine workings, but only in the developed parts of the deposit. The present study considers selected seams of two mines. The methodology used is based on detailed geostatistical analysis, point kriging procedure and P. Gy’s theory of sampling. Its purpose is: • defining and comparing geostatistical models for variability of the ash and sulphur contents for data originating from boreholes and mine workings, • predicting by means of point kriging the values of the parameters and errors of interpolation using data from boreholes at grid points where underground mine workings were later channel-sampled, • assessing the accuracy of interpolation by comparison of predicted values of parameters with real values (found by channel sampling), • evaluating the variances of total secondary sampling error (error of preparation of assay samples) and analytical error introduced by assaying of sulphur and ash, • assessing the contribution of sampling and analytical errors (global estimation error) to the interpolation errors. The authors found that the interpolation errors for ash or sulphur content are very large, with mean relative values of 35%-60%, mainly caused by the considerable natural variability, a significant role of random component of variability, and heterogeneity of spatial distribution of these characteristics. The sampling and analytical errors play a negligible role. Their values are smaller than 11% of interpolation error values. Presenting estimates of the spatial distribution of ash and sulphur contents in coal seams by means of contour maps is unreasonable if they are based on drill core sampling.

Distributed state estimation for linear multi-agent systems with time-varying measurement topology

This paper addresses the problem of distributed state estimation in formations of vehicles with time-varying measurement topology. One or more vehicles have access to measurements of their own state, while the rest must rely on measurements relative to other vehicles in the vicinity and limited communication to estimate their own state, and it is assumed that the vehicles may gain or lose access to some measurements over time. The proposed solution consists in the implementation of a local state observer on-board each vehicle, and the effects of changes in the measurement topology on the estimation error dynamics are studied resorting to switched systems theory. Sufficient conditions for exponential stability of the global estimation error dynamics are presented for two different switching laws. The results are applied to the practical case of a formation of Autonomous Underwater Vehicles (AUVs), and simulation results are presented that validate the proposed solution.