Residual Based Error Indicator Research Articles

A priori and a posteriori error estimates of a space–time Petrov–Galerkin spectral method for time-fractional diffusion equation

Time-fractional diffusion equation is an important transport dynamical model for simulating fractal time random walk. This article is devoted to investigating the a priori and a posteriori error estimates for this model equation. A Petrov–Galerkin spectral method is revisited in this paper to address our problem, which the generalized Jacobi functions and Fourier-like basis functions are utilized as basis for constructing efficient and accurate space–time spectral approximations. Rigorous proofs are given for the stability of our spectral scheme. And then the convergence of the proposed method is proved by establishing an a priori error estimate. Specifically, an efficient and reliable a posteriori error estimator is introduced, and we derive that the residual-based error indicator provides an upper bound and a lower bound for the numerical error. Finally, several numerical experiments are provided to examine our theoretical claims.

GLaSDI: Parametric physics-informed greedy latent space dynamics identification

A parametric adaptive physics-informed greedy Latent Space Dynamics Identification (gLaSDI) method is proposed for accurate, efficient, and robust data-driven reduced-order modeling of high-dimensional nonlinear dynamical systems. In the proposed gLaSDI framework, an autoencoder discovers intrinsic nonlinear latent representations of high-dimensional data, while dynamics identification (DI) models capture local latent-space dynamics. An interactive training algorithm is adopted for the autoencoder and local DI models, which enables identification of simple latent-space dynamics and enhances accuracy and efficiency of data-driven reduced-order modeling. To maximize and accelerate the exploration of the parameter space for the optimal model performance, an adaptive greedy sampling algorithm integrated with a physics-informed residual-based error indicator and random-subset evaluation is introduced to search for the optimal training samples on the fly. Further, to exploit local latent-space dynamics captured by the local DI models for an improved modeling accuracy with a minimum number of local DI models in the parameter space, a k-nearest neighbor convex interpolation scheme is employed. The effectiveness of the proposed framework is demonstrated by modeling various nonlinear dynamical problems, including Burgers equations, nonlinear heat conduction, and radial advection. The proposed adaptive greedy sampling outperforms the conventional predefined uniform sampling in terms of accuracy. Compared with the high-fidelity models, gLaSDI achieves 17 to 2,658× speed-up with 1 to 5% relative errors.

An interface preserving and residual-based adaptivity for phase-field modeling of fully Eulerian fluid-structure interaction

In this paper, we present a novel interface preserving and residual-based adaptivity procedure for the phase-field modeling of a fully Eulerian fluid-structure interaction. The proposed adaptive fully Eulerian procedure involves a fixed background unstructured finite element mesh on which the fluid-structure boundary is treated implicitly via a diffused interface description. We employ the stabilized finite element formulation and a partitioned iterative procedure to solve the coupled system of the Allen-Cahn phase-field equation with the unified momentum equation comprising solid and fluid dynamics. To evaluate the solid stresses, the left Cauchy-Green deformation tensor is convected at each time step to trace the evolution of the solid strain in the Eulerian reference frame. We utilize a residual based error indicator and the newest vertex bisection algorithm for the adaptive refinement/coarsening of the unstructured mesh. Through the adaptive finite element method, we aim to preserve the profile of the diffused interface in the presence of convective distortion. We propose an early stopping criterion for the adaptivity of the interface dynamics by combining the effects of the phase-field interface thickness parameter and the mesh resolution. In addition, we control the size of the refined/coarsened elements by introducing a characteristic length scale governed by a weighted harmonic mean of the desired bulk and the interface mesh resolutions. The proposed stopping criterion and the characteristic length scale provide significant reductions in the computational cost while simultaneously ensuring convergence properties of the coupled fluid-structure system. We perform a detailed convergence and accuracy analysis via two benchmark problems namely, the pure solid system and a coupled fluid-solid system with an interface in a rectangular domain. We further provide a comparative study between feature- and residual based adaptive strategies. We systematically assess the performance of the adaptive procedure in terms of conservation properties and computational efficiency. Finally, we demonstrate our fully Eulerian adaptive solver to simulate the contact and bouncing phenomenon between an elastic solid and a rigid wall.

A posteriori error estimates for Biot system using a mixed discretization for flow

We first derive convergence and a priori stability, and next reliability and efficiency of a posteriori error indicators for a Biot poroelastic model coupled with an elastic model in R3, solved by a continuous Galerkin scheme (CG) for the displacement and a mixed finite element scheme for the flow. The coupled system is decoupled by a fixed stress splitting algorithm. The numerical implementation of the residual based error indicators is simple, even for the mixed discretization, but at the expense of two suboptimal bounds. The scheme is tested on two benchmark problems via several numerical experiments.

A novel adaptive mesh free approach for even- parity neutron transport equation

In this paper, a novel spatially adaptive Mesh free approach using local nodes and radial basis functions, based on weak-form formulation is presented to solve the even-parity neutron transport equation. A residual based error indicator is proposed and used in an adaptive scheme for error estimation. The local balance of neutrons is used as an indicator to determine the appropriate regions of problem for node refinement. In this regard our previous computer program, MFreent, is upgraded for solving neutron transport equation using K+ variational principle in an adaptive scheme for two-dimensional X-Y geometry. The multi-quadrics radial basis function is used to construct the radial point interpolation shape functions for spatial approximation of even-parity angular flux. Also, the angular part of flux is implemented via spherical harmonics expansion. Some numerical test cases are investigated to evaluate the performance of the locally adaptive node-refinement approach for both fixed source and eigenvalue problems solution. The obtained results confirm the solution accuracy and efficiency of the proposed method.

Glasdi: Parametric Physics-Informed Greedy Latent Space Dynamics Identification

A parametric adaptive physics-informed greedy Latent Space Dynamics Identification (gLaSDI) method is proposed for accurate, efficient, and robust data-driven reduced-order modeling of high-dimensional nonlinear dynamical systems. In the proposed gLaSDI framework, an autoencoder discovers intrinsic nonlinear latent representations of high-dimensional data, while dynamics identification (DI) models capture local latent-space dynamics. An interactive training algorithm is adopted for the autoencoder and local DI models, which enables identification of simple latent-space dynamics and enhances accuracy and efficiency of data-driven reduced-order modeling. To maximize and accelerate the exploration of the parameter space for the optimal model performance, an adaptive greedy sampling algorithm integrated with a physics-informed residual-based error indicator and random-subset evaluation is introduced to search for the optimal training samples on-the-fly. Further, to exploit local latent-space dynamics captured by the local DI models for an improved modeling accuracy with a minimum number of local DI models in the parameter space, an efficient k-nearest neighbor convex interpolation scheme is employed. The effectiveness of the proposed framework is demonstrated by modeling various nonlinear dynamical problems, including Burgers equations, nonlinear heat conduction, and radial advection. The proposed adaptive greedy sampling outperforms the conventional predefined uniform sampling in terms of accuracy. Compared with the high-fidelity models, gLaSDI achieves 66 to 4,417x speed-up with 1 to 5% relative errors.

A ROM-accelerated parallel-in-time preconditioner for solving all-at-once systems in unsteady convection-diffusion PDEs

In this paper we propose a model reduction technique to speed up the diagonalization-based parallel-in-time (ParaDIAG) preconditioner, for iteratively solving all-at-once systems from evolutionary PDEs. In particular, we use the reduced basis method to seek a low-dimensional approximation to the sequence of complex-shifted systems arising from Step-(b) of the ParaDIAG preconditioning procedure. Different from the standard reduced order modeling that uses the separation of offline and online stages, we have to build the reduced order model (ROM) online for the considered systems at each iteration. Therefore, several heuristic acceleration techniques are introduced in the greedy basis generation algorithm, that is built upon a residual-based error indicator, to further boost up its computational efficiency. Several numerical experiments are conducted, which illustrate the favorable computational efficiency of our proposed ROM-accelerated ParaDIAG preconditioner, in comparison with the multigrid-based one.

Adaptive analysis using scaled boundary finite element method in 3D

In this paper, an adaptive refinement technique using the scaled boundary finite element method (SBFEM) is proposed. The salient feature of this technique is that it is not required to regenerate the mesh for the whole model during the iterations. To this end, a local mesh refinement strategy is implemented based on a polytree algorithm in three dimensions, which can be applied to polyhedral elements with arbitrary number of nodes, edges and faces. These elements constructed by the SBFEM can be used in analysis with their boundaries discretized only, which reduce the difficulty to connect elements with different sizes. An explicit residual based error indicator is developed using the discontinuity of the stress field to guide the adaptive mesh refinement. The accuracy and efficiency of the proposed method are demonstrated using five numerical examples, including complex geometry and stress singularity.

A robust error estimator and a residual-free error indicator for reduced basis methods

The Reduced Basis Method (RBM) is a rigorous model reduction approach for solving parameterizedpartial differential equations. It identifies a low-dimensional subspace for approximation of the parametric solution manifold that is embedded in high-dimensional space. A reduced order model is subsequently constructed in this subspace. RBM relies on residual-based error indicators or a posteriori error bounds to guide construction of the reduced solution subspace, to serve as a stopping criteria, and to certify the resulting surrogate solutions. Unfortunately, it is well-known that the standard algorithm for residual norm computation suffers from premature stagnation at the level of the square root of machine precision.In this paper, we develop two alternatives to the standard offline phase of reduced basis algorithms. First, we design a robust strategy for computation of residual error indicators that allows RBM algorithms to enrich the solution subspace with accuracy beyond root machine precision. Secondly, we propose a new error indicator based on the Lebesgue function in interpolation theory. This error indicator does not require computation of residual norms, and instead only requires the ability to compute the RBM solution. This residual-free indicator is rigorous in that it bounds the error committed by the RBM approximation, but up to an uncomputable multiplicative constant. Because of this, the residual-free indicator is effective in choosing snapshots during the offline RBM phase, but cannot currently be used to certify error that the approximation commits. However, it circumvents the need for a posteriori analysis of numerical methods, and therefore can be effective on problems where such a rigorous estimate is hard to derive.

Adaptive Least-Squares Mixed Generalized Multiscale Finite Element Methods

In this paper, we present two kinds of adaptive least-squares mixed generalized multiscale finite element methods (GMsFEMs) for solving an elliptic problem in highly heterogeneous porous media. An offline adaptive method is developed through iteratively enriching the local velocity and pressure multiscale basis functions based on residual-based error indicators. In addition, an online adaptive method is also proposed to create the new multiscale basis functions in the online stage for both velocity and pressure. The enriched basis functions are computed by the residual and maximizing the reduction in error. The offline adaptive method attempts to provide a good approximation space based on the heterogeneity of the coefficient, and the online adaptive method aims at providing a good approximation space by taking the given source into account. Both of the adaptive methods can achieve a better approximation than the uniform enrichment method using the same number of basis functions. Convergence analysis is carried out for the adaptive least-squares GMsFEMs. The analysis suggests that, by choosing a suitable number of initial basis functions for velocity and pressure, the online adaptive method can render a faster convergence rate compared with the offline adaptive method and the uniform enrichment. A few numerical results are presented to confirm the analysis and the performance of the presented adaptive multiscale methods.

Adaptive large eddy simulation

Several a posteriori indicators in the framework of local grid adaptation and large eddy simulation (LES) are evaluated. In LES indicators must be capable to bound not only the discretisation error, but also the modeling error. Moreover, the numerical method must be able to adapt the computational grid dynamically, as the regions requiring different resolution are not static. The performance of different indicators is evaluated in two flow configurations. It turns out that the classic residual based error indicator and the newly introduced heuristic indicator perform best.

Adaptive discontinuous Galerkin methods for state constrained optimal control problems governed by convection diffusion equations

We study a posteriori error estimates for the numerical approximations of state constrained optimal control problems governed by convection diffusion equations, regularized by Moreau---Yosida and Lavrentiev-based techniques. The upwind Symmetric Interior Penalty Galerkin (SIPG) method is used as a discontinuous Galerkin (DG) discretization method. We derive different residual-based error indicators for each regularization technique due to the regularity issues. An adaptive mesh refinement indicated by a posteriori error estimates is applied. Numerical examples are presented to illustrate the effectiveness of the adaptivity for both regularization techniques.

Efficient simulation of cardiac electrical propagation using high-order finite elements II: Adaptive p-version

We present a computationally efficient method of simulating cardiac electrical propagation using an adaptive high-order finite element method to automatically concentrate computational effort where it is most needed in space on each time-step. We drive the adaptivity using a residual-based error indicator, and demonstrate using norms of the error that the indicator allows us to control it successfully.Our results using two-dimensional domains of varying complexity demonstrate that significant improvements in efficiency are possible over the standard linear FEM in our single-thread studies, and our preliminary three-dimensional results suggest that improvements are also possible in 3D. We do not work in parallel or investigate the challenges for adaptivity such as dynamic load-balancing which are associated with parallelisation. However, based upon recent work demonstrating that in some circumstances and with moderate processor counts parallel h-adaptive methods are efficient, and upon the claim that p-adaptivity will outperform h-adaptivity, we argue that p-adaptivity should be investigated for efficiency in parallel for simulation on moderate numbers of processors.

A Simple and Efficient Isotropic Mesh Adaptation Scheme Based on Control-Volume Flux Imbalance

The primary goal of this work is to develop a simple residual-based error indicator in the context of the finite-volume method that is as easy to use as conventional gradient-based error indicators and as robust as residual-based error indicators. The residual is defined as the cell imbalance error or rather the net flow error across the faces of each control-volume. To estimate the residual, the neglected Taylor series terms in the discretization of the face flows are used. Then a simplified residual-based error indicator suitable for isotropic triangular meshes is developed. A mesh adaptation scheme is used to minimize the global value of the error indicator through local refinement, coarsening, and vertex repositioning. The adaptation results for the lid-driven cavity flow at Re = 1,600 and the impinging slot jet flow at Re = 500 confirm that the proposed adaptation scheme can effectively decrease the discretization error by an order of magnitude in fewer than 10 iterations.

Numerical investigation of effectivity indices of space-time error indicators for Navier–Stokes equations

In this paper we propose two error indicators aimed at estimating the space discretization error and the time discretization error for the unsteady Navier–Stokes equations. We define a space error indicator for evaluating the quality of the mesh and a time error indicator for evaluating the time discretization error. Moreover, we verify the reliability of the estimations through numerical experiments and we propose an effective space-time adaptive strategy for the unsteady Navier–Stokes equations. Such technique is based on two residual-based error indicators that suitably drive the mesh and the timestep-length modifications. Adaptive simulations show that the presented strategy allows to obtain accurate solutions in efficient way.

A gradient smoothing method (GSM) based on strong form governing equation for adaptive analysis of solid mechanics problems

A gradient smoothing method (GSM) based on strong form of governing equations for solid mechanics problems is proposed in this paper, in which gradient smoothing technique is used successively over the relevant gradient smoothing domains to develop the first- and second-order derivative approximations by calculating weights for a set of field nodes surrounding a node of interest. The GSM is found very stable and can be easily applied to arbitrarily irregular triangular meshes for complex geometry. Unlike other strong form methods, the present method has excellent stability that is crucial for adaptive analysis. An effective and robust residual based error indicator and simple refinement procedure using Delaunay diagram are then implemented in our GSM for adaptive analyses. The reliability and performance of the proposed GSM for adaptive procedure are demonstrated in several solid mechanics problems including problems with singularities and concentrated loading, compared with the well-known finite element method (FEM).

On efficient simulation of non-Newtonian flow in saturated porous media with a multigrid adaptive refinement solver

Flow of non-Newtonian fluid in saturated porous media can be described by the continuity equation and the generalized Darcy law. Here we discuss the efficient solution of the resulting second order nonlinear elliptic equation. The equation is discretized by the finite volume method on a cell-centered grid. Local adaptive refinement of the grid is introduced in order to reduce the number of unknowns. We develop a special implementation, that allows us to perform unstructured local refinement in conjunction with the finite volume discretization. Two residual based error indicators are exploited in the adaptive refinement criterion. Second order accurate discretization of the fluxes on the interfaces between refined and non-refined subdomains, as well as on the boundaries with Dirichlet boundary condition, are presented here as an essential part of an accurate and efficient algorithm. A nonlinear full approximation storage multigrid algorithm is developed especially for the above described composite (coarse plus locally refined) grid approach. In particular, second order approximation of the fluxes around interfaces is a result of a quadratic approximation of slave nodes in the multigrid-adaptive refinement (MG-AR) algorithm. Results from numerical solution of various academic and practice-induced problems are presented and the performance of the solver is discussed.

A regularized least-squares radial point collocation method (RLS-RPCM) for adaptive analysis

This paper presents a stabilized meshfree method formulated based on the strong formulation and local approximation using radial basis functions (RBFs). The purpose of this paper is two folds. First, a regularization procedure is developed for stabilizing the solution of the radial point collocation method (RPCM). Second, an adaptive scheme using the stabilized RPCM and residual based error indicator is established. It has been shown in this paper that the features of the meshfree strong-form method can facilitated an easier implementation of adaptive analysis. A new error indicator based on the residual is devised and used in this work. As shown in the numerical examples, the new error indicator can reflect the quality of the local approximation and the global accuracy of the solution. A number of examples have been presented to demonstrate the effectiveness of the present method for adaptive analysis.

A posteriori discontinuous Galerkin error estimates for transient convection–diffusion equations

A posteriori error estimates are derived for unsteady convection–diffusion equations discretized with the non-symmetric interior penalty and the local discontinuous Galerkin methods. First, an error representation formula in a user specified output functional is derived using duality techniques. Then, an L t 2 ( L x 2 ) a posteriori estimate consisting of elementwise residual-based error indicators is obtained by eliminating the dual solution. Numerical experiments are performed to assess the convergence rates of the various error indicators on a model problem.

A Quasi-Static Condition for Enhancing p-Adaptive, Mixed-Order Element, FE Analysis

This paper relates to p-adaptive algorithms for the three-dimensional, full-wave,time-harmonic, electric field Galerkin procedure finite element method, using mixed- and/or complete-order elements. A quasi-static electric field condition for enhancing the performance of such algorithms is proposed. The condition is used to distinguish between mixed- and complete-order upgrades. A p-adaptive algorithm, driven by an explicit, residual-based error indicator, is presented. Within this algorithm, the effectiveness of the quasi-static condition is evaluated numerically by considering three common microwave engineering structures.