A-posteriori Error Estimator Research Articles

A-priori and a-posteriori error estimates for discontinuous Galerkin method of the Maxwell eigenvalue problem

This paper is devoted to a-priori and a-posteriori error analysis of discontinuous Galerkin (DG) method for the Maxwell eigenvalue problem. The discrete compactness of DG space is proved so that the Babuška and Osborn spectral approximation theory can be applicable in the a-priori error analysis. Then we prove the optimal error estimates for DG eigenfunctions in mesh-dependent norm and DG eigenvalues. A special contribution of this work is to prove that the error in L2-norm for smooth eigenfunctions is of higher order than that in mesh-dependent norm, so that the DG eigenvalues can approximate the true eigenvalues from upper. Another contribution of this work is to provide a-posteriori error analysis for the DG method. A reliable a-posteriori error estimator is analyzed. The upper bound property of DG eigenvalues and the robustness of adaptive methods are verified through numerical experiments.

An adaptive certified space-time reduced basis method for nonsmooth parabolic partial differential equations

In this paper, a nonsmooth semilinear parabolic partial differential equation (PDE) is considered. For a reduced basis (RB) approach, a space-time formulation is used to develop a certified a-posteriori error estimator. This error estimator is adopted to the presence of the discrete empirical interpolation method (DEIM) as approximation technique for the nonsmoothness. The separability of the estimated error into an RB and a DEIM part then guides the development of an adaptive RB-DEIM algorithm, combining both offline phases into one. Numerical experiments show the capabilities of this novel approach in comparison with classical RB and RB-DEIM approaches.

An Adaptive PML Finite Volume Algorithm for the Scattering by Periodic Gratings

In this paper, we studied the finite volume formulations for solving the diffraction grating problem that is truncated by the perfectly matched layer (PML) technique. Based on a reliable the a-posteriori error estimate, an adaptive PML finite volume method is discussed for the numerical approximation of the diffraction grating problem. The PML parameters are obtained numerically by sharp a-posteriori error estimates of the PML finite volume method such as the thickness of the layer and the medium property. It is worth mentioning in the a-posteriori error estimates that we derive the error representation formula and use a [Formula: see text]-orthogonality property of the residual similar to the Galerkin orthogonality used in the finite element method. Furthermore, the lower bound is established to demonstrate the efficiency of the a-posteriori error estimates. Numerical experiments are given to illustrate the accuracy and robustness of our adaptive algorithm.

Reduced Basis modelling of turbulence with well-developed inertial range

In this work, we introduce a Reduced Basis model for turbulence at statistical equilibrium. This is based upon an a-posteriori error estimation procedure that measures the distance from a trial solution to the K41 theory energy spectrum. We apply this general idea to build a Reduced Basis Smagorinsky turbulence model through a Greedy Algorithm. We derive some error estimates that make apparent the role of the energy spectrum in the ROM approximation. We carry on some tests for some academic unsteady 2D flows at large Reynolds number, that present well developed inertial spectrum. The methods presents a high efficiency, as the error achieved with the reduced method is 3 to 4 times the ones achieved if the exact error is used in the Greedy Algorithm.

A Simple Strain Recovery based A-posteriori Error Estimator for Laminated Composite Plates

A globally and locally reliable strain recovery based a-posteriori error estimator for laminated composite plates is proposed in this study. An extensive check of the local and global quality of the proposed error estimator is carried out for the bending problem. Effect of material orientation, ply stacking sequence, boundary condition and loading profile is carried out for a square plate. It is found that this estimator is reliable for almost all the cases, and even predicts errors for the preasymptotic range reliably.

Pointwise-in-time a posteriori error control for higher-order discretizations of time-fractional parabolic equations

Time-fractional parabolic equations with a Caputo time derivative are considered. For such equations, we explore and further develop the new methodology of the a-posteriori error estimation and adaptive time stepping proposed in Kopteva (2022). We improve the earlier time stepping algorithm based on this theory, and specifically address its stable and efficient implementation in the context of high-order methods. The considered methods include an L1-2 method and continuous collocation methods of arbitrary order, for which adaptive temporal meshes are shown to yield optimal convergence rates in the presence of solution singularities.

A robust and accurate adaptive approximation method for a diffuse-interface model of binary-fluid flows

We present an adaptive simulation framework for binary-fluid flows, based on the Abels–Garcke–Grün Navier–Stokes–Cahn–Hilliard (AGG NSCH) diffuse-interface model. The adaptive-refinement procedure is guided by a two-level hierarchical a-posteriori error estimate, and it effectively resolves the spatial multiscale behavior of the diffuse-interface model. To improve the robustness of the solution procedure and avoid severe time-step restrictions for small-interface thicknesses, we introduce an ɛ-continuation procedure, in which the diffuse interface thickness (ɛ) are enlarged on coarse meshes, and the mobility is scaled accordingly. To further accelerate the computations and improve robustness, we apply a modified Backward Euler scheme in the initial stages of the adaptive-refinement procedure in each time step, and a Crank–Nicolson scheme in the final stages of the refinement procedure. To enhance the robustness of the nonlinear solution procedure, we introduce a partitioned solution procedure for the linear tangent problems in Newton’s method, based on a decomposition of the NSCH system into its NS and CH subsystems. We conduct a systematic investigation of the conditioning of the monolithic NSCH tangent matrix and of its NS and CH subsystems for a representative 2D model problem. To illustrate the properties of the presented adaptive simulation framework, we present numerical results for a 2D oscillating water droplet suspended in air, and we validate the obtained results versus those of a corresponding sharp-interface model.

Guaranteed a-posteriori error estimation for semi-discrete solutions of parabolic problems based on elliptic reconstruction

This paper addresses the construction of error estimate for semi-discrete solutions of parabolic problems based on the superconvergence properties in the elliptic reconstruction.

A multi-criteria h-adaptive finite-element framework for industrial part-scale thermal analysis in additive manufacturing processes

This work presents an h-adaptive finite-element (FE) strategy to address the numerical simulation of additive manufacturing (AM) of large-scale parts. The wire-arc additive manufacturing is chosen as the demonstrative technology for its manufacturing capabilities suitable for industrial purposes. The scanning path and processing parameters of the simulation are provided via a RS-274 (GCode) file, being the same as the one delivered to the AM machine. The approach is suitable for industrial applications and can be applied to other AM processes. To identify the location in the FE mesh of the heat affected zone (HAZ), a collision detection algorithm based on the separating axis theorem is used. The mesh is continuously adapted to guarantee the necessary mesh resolution to capture the phenomena inside and outside the HAZ. To do so, a multi-criteria adaptive mesh refinement and coarsening (AMR) strategy is used. The AMR includes a geometrical criterion to guarantee the FE size within the HAZ, and a Zienkiewicz–Zhu-based a-posteriori error estimator to guarantee the solution accuracy elsewhere. Thus, the number of active FEs is controlled and mesh manipulation by the end-user is avoided. Numerical simulations comparing the h-adaptive strategy with the (reference) fixed fine meshes are performed to prove the computational cost efficiency and the solution accuracy.

On an accurate A-posteriori error estimator and adaptive time stepping for the implicit and explicit composite time integration algorithms

This paper focuses on the design of an accurate and versatile A-posteriori error estimation and adaptive time stepping for general composite schemes and typical of the implicit ρ∞-Bathe and explicit Noh-Bathe composite methods for time integration in second-order transient systems. Two novel error estimators, Disp/P3 and Disp/P4, are newly proposed based on the generalized polynomials. They have generalized compact single-step representations and work for both implicit and explicit, dissipative and non-dissipative composite schemes. Validation tests of linear and nonlinear problems show that the estimated local error reaches excellent agreement with the exact local error, and the third-order convergence rates of the local error are obtained. Besides, the Disp/P4 is more accurate than Disp/P3 as the effectivity index is almost identical to unity when random time steps are taken into consideration, and hence the Disp/P4 is highly recommended to be merged to the adaptive time stepping procedure. Three examples encompassing the wave propagation, damped/undamped spring-pendulum, and nonlinear spring-mass system are solved by implicit and explicit composite methods via the adaptive time stepping. Numerical results show the advantage of adaptive time stepping based on the proposed error estimator in contrast to constant time stepping regarding the CPU cost.

An A Posteriori Error Estimate of the Outer Normal Derivative Using Dual Weights

We derive a residual based a-posteriori error estimate for the outer normal flux of approximations to {the diffusion problem with variable coefficient}. By analyzing the solution of the adjoint problem, we show that error indicators in the bulk may be defined to be of higher order than those close to the boundary, which lead to more economic meshes. The theory is illustrated with some numerical examples.

A robust and scalable unfitted adaptive finite element framework for nonlinear solid mechanics

In this work, we bridge standard Adaptive Mesh Refinement and coarsening (AMR) on scalable octree background meshes and robust unfitted Finite Element (FE) formulations for the automatic and efficient solution of large-scale nonlinear solid mechanics problems posed on complex geometries, as an alternative to standard body-fitted formulations, unstructured mesh generation and graph partitioning strategies. We pay special attention to those aspects requiring a specialized treatment in the extension of the unfitted h-adaptive Aggregated Finite Element Method (h-AgFEM) on parallel tree-based adaptive meshes, recently developed for linear scalar elliptic problems, to handle nonlinear problems in solid mechanics. In order to accurately and efficiently capture localized phenomena that frequently occur in nonlinear solid mechanics problems, we perform pseudo time-stepping in combination with h-adaptive dynamic mesh refinement and re-balancing driven by a-posteriori error estimators. The method is implemented considering both irreducible and mixed (u/p) formulations and thus it is able to robustly face problems involving incompressible materials. In the numerical experiments, both formulations are used to model the inelastic behavior of a wide range of compressible and incompressible materials. First, a selected set of benchmarks is reproduced as a verification step. Second, a set of experiments is presented with problems involving complex geometries. Among them, we model a cantilever beam problem with spherical hollows distributed in a simple cubic (SC) array. This test involves a discrete domain with up to 11.7M Degrees Of Freedom (DOFs) solved in less than two hours on 3072 cores of a parallel supercomputer.

A-posteriori reduced basis error-estimates for a semi-discrete in space quasilinear parabolic PDE

We prove a-posteriori error-estimates for reduced-order modeling of quasilinear parabolic PDEs with non-monotone nonlinearity. We consider the solution of a semi-discrete in space equation as reference, and therefore incorporate reduced basis-, empirical interpolation-, and time-discretization-errors in our consideration. Numerical experiments illustrate our results.

A-posteriori error estimations based on postprocessing technique for two-sided fractional differential equations

The analysis of diffusion-reaction equations with general two-sided fractional derivative characterized by a parameter p∈[0,1] is investigated in this paper. First, we present a Petrov-Galerkin method, derive a proper weak formulation and show the well-posedness of its weak solution. Moreover, on the basis of the two-sided Jacobi polyfractonomials, a priori error analysis of Petrov-Galerkin method is derived. Further, a posteriori error analysis is established rigorously. More precisely, we develop a novel postprocessing technique to enhance the Petrov-Galerkin method by adding a small amount of computation, and analyze asymptotically exact a-posteriori error estimators. Finally, we demonstrate the theoretical results with numerical examples.

Space–time least-squares finite elements for parabolic equations

We present a space–time least-squares finite element method for the heat equation. It is based on residual minimization in L2 norms in space–time of an equivalent first order system. This implies that (i) the resulting bilinear form is symmetric and coercive and hence any conforming discretization is uniformly stable, (ii) stiffness matrices are symmetric, positive definite, and sparse, (iii) we have a local a-posteriori error estimator for free. In particular, our approach features full space–time adaptivity. We also present a-priori error analysis on simplicial space–time meshes which are highly structured. Numerical results conclude this work.

A-posteriori error estimations of the GJF-Petrov–Galerkin methods for fractional differential equations

In this paper, we develop an efficient spectral GJF-Petrov–Galerkin algorithm and the postprocessed method to solve a class of fractional initial value problems. The main part of these algorithms is to use a special set of general Jacobi functions (GJFs) to form the trial space and test space. We give a rigorous error analysis in non-uniformly weighted Sobolev spaces and obtain optimal error estimates. In particular, the postprocessing technique is used to construct the postprocessed method. In addition, we derive its superconvergence estimates and define a-posteriori error estimators that are asymptotically exact. Numerical experiments are included to support the theoretical analysis.

An adaptive least-squares finite element method for Giesekus viscoelastic flow problems

In this study, a least-squares (LS) finite element method with an adaptive mesh approach is investigated for Giesekus viscoelastic flow problems. We consider the weighted LS method on uniform and adaptive meshes for the Newton linearized viscoelastic problem, where adaptive grids are automatically generated by the least-squares solutions. We use a residual-type a-posteriori error estimator to adjust weights in the LS functional and compare the convergence behaviour of adaptive meshes generated using different grading functions. Numerical results demonstrate that the adaptive LS method shows at least the first-order convergence rate when equal-order linear interpolation functions are used for all variables, which agrees with the theoretical estimate. In addition, adaptive grids generated using the velocity outperform those based on the a-posteriori error estimator, yielding better numerical results.

Solution of the three-dimensional inverse elastography problem for parametric classes of inclusions

We study the three-dimensional inverse problem of elastography, that is finding the Young's modulus of a biological tissue from known values of its vertical displacements. In this way, one can find inclusions with Young's modulus several times higher than its known background value. Such inclusions are interpreted as tumours. A quasistatic statement of the problem is used in which the fragment of the tissue is considered as a linearly elastic body. It is assumed that the geometry of inclusions is specified parametrically, and the Young's modulus inside and outside the inclusions is constant. The task is reduced to finding the number of inclusions, parameters defining their shape and the Young's modulus inside the inclusions. To solve the problem, a special algorithm is proposed and justified. The results of numerical experiments on solving a three-dimensional model problems are presented. A comparison is made of the solutions to the inverse problem in the three-dimensional domain and two-dimensional inverse problems in selected cross-sections of this domain. It is found that 2D inverse problems do not always allow one to find a true 3D solution. For one of solved inverse problems, a-posteriori error estimates for Young modulus and geometric parameters of inclusions are obtained.

Local adaptive [formula omitted]-enrichments and generalized finite elements for transient heat diffusion problems

A residual a-posteriori error estimate is used in conjunction with a q−adaptive procedure for selecting enrichment functions in the modelling of transient heat diffusion problems with the Generalized Finite Element Method (GFEM). The error estimate allows to assess local as well as global errors of the GFEM solutions and provides a tool to adaptively enrich the approximation space. With the proposed adaptive algorithm, the total number of degrees of freedom (DOFs) is significantly reduced in comparison to uniform enrichment. For a comparable accuracy, a reduction of up to 80% in DOFs is achieved. Moreover, the error estimate captures the deterioration of the conditioning of the system matrix typically encountered in enrichment based methods and enables the possibility of keeping the condition number within an acceptable limit.

Building a certified reduced basis for coupled thermo-hydro-mechanical systems with goal-oriented error estimation

A goal-oriented a-posteriori error estimator is developed for transient coupled thermo-hydro-mechanical (THM) parametric problems solved with a reduced basis approximation. The estimator assesses the error in some specific Quantity of Interest (QoI). The goal-oriented error estimate is derived based on explicitly-computed weak residual of the primal problem and implicitly-computed adjoint solution associated with the QoI. The time-dependence of the coupled THM system poses an additional complexity as the auxiliary adjoint problem evolves backwards in time. The error estimator guides a greedy adaptive procedure that constructs progressively an optimal reduced basis by smartly selecting snapshot points over a given parametric training sample. The reduced basis obtained is used to drastically reduce the coupled system spatial degrees of freedom by several orders of magnitude. The computational gain obtained from the developed methodology is demonstrated through applications in 2D and 3D parametrized problems simulating the evolution of coupled THM processes in rock masses.