Residual-based Error Estimator Research Articles

Functional approach to the error control in adaptive IgA schemes for elliptic boundary value problems

This work presents a numerical study of functional type a posteriori error estimates for IgA approximation schemes in the context of elliptic boundary-value problems. Along with the detailed discussion of the most crucial properties of such estimates, we present the algorithm of a reliable solution approximation together with the scheme of an efficient a posteriori error bound generation. In this approach, we take advantage of B-(THB-) spline’s high smoothness for the auxiliary vector function reconstruction, which, at the same time, allows to use much coarser meshes and decrease the number of unknowns substantially. The most representative numerical results, obtained during a systematic testing of error estimates, are presented in the second part of the paper. The efficiency of the obtained error bounds is analysed from both the error estimation (indication) and the computational expenses points of view. Several examples illustrate that functional error estimates (alternatively referred to as the majorants and minorants of deviation from an exact solution) perform a much sharper error control than, for instance, residual-based error estimates. Simultaneously, assembling and solving routines for an auxiliary variables reconstruction, which generate the majorant (or minorant) of an error, can be executed several times faster than the routines for a primal unknown.

An adaptive variational procedure for the conservative and positivity preserving Allen–Cahn phase-field model

In this paper, we present an adaptive variational procedure for unstructured meshes to capture fluid–fluid interfaces in two-phase flows. The two phases are modeled by the phase-field finite element formulation, which involves the conservative Allen–Cahn equation coupled with the incompressible Navier–Stokes equations. The positivity preserving variational formulation is designed to maintain the bounded and stable solution of the Allen–Cahn equation. For the adaptivity procedure, we consider the residual-based error estimates for the underlying differential equations of the two-phase system. In particular, the adaptive refinement/coarsening is carried out by the newest vertex bisection algorithm by evaluating the residual error indicators based on the error estimates of the Allen–Cahn equation. The coarsening algorithm avoids the storage of the tree data structures for the hierarchical mesh, thus providing the ease of numerical implementation. Furthermore, the proposed nonlinear adaptive partitioned procedure aims at reducing the amount of coarsening while maintaining the convergence properties of the underlying nonlinear coupled differential equations. We investigate the adaptive phase-field finite element scheme through the spinodal decomposition in a complex curved geometry and the volume-conserved interface motion driven by the mean curvature flow for two circles in a square domain. We then assess the accuracy and efficiency of the proposed procedure by modeling the free-surface motion in a sloshing tank. In contrast to the non-adaptive Eulerian grid counterpart, we demonstrate that the mesh adaptivity remarkably reduces the degrees of freedom and the computational cost by nearly half for similar accuracy. The mass loss in the Allen–Cahn equation via adaptivity process is also reduced by nearly three times compared to the non-adaptive mesh. Finally, we apply the adaptive numerical framework to solve the application of a dam-breaking problem with topological changes.

Adaptive discontinuous Galerkin approximation of optimal control problems governed by transient convection-diffusion equations

In this paper, we investigate a posteriori error estimates of a control-constrained optimal control problem governed by a time-dependent convection diffusion equation. The control constraints are handled by using the primal-dual active set algorithm as a semi-smooth Newton method and by adding a Moreau-Yosida-type penalty function to the cost functional. Residual-based error estimators are proposed for both approaches. The derived error estimators are used as error indicators to guide the mesh refinements. A symmetric interior penalty Galerkin method in space and a backward Euler method in time are applied in order to discretize the optimization problem. Numerical results are presented, which illustrate the performance of the proposed error estimators.

A residual-based h-adaptive reconstructed discontinuous Galerkin method for the compressible Euler equations on unstructured grids

In this paper, a residual-based h-adaptive high-order reconstructed DG (rDG) method based on the hybrid reconstruction strategy is developed for solving the compressible Euler equations on unstructured grids. The proposed method combines the advantages of high-order rDG discretization with appropriate residual-based error estimation techniques and h-adaptive refinement strategies, which exhibits its superior potential compared to the underlying DG methods. To be specific, a third-order hybrid rDG(p1p2) method has been carefully designed and evaluated on incompatible quadrilateral grids with hanging nodes in order to preserve 2-exactness property during the implementation of mesh refinement and coarsening. A residual-based error estimator is used as local error indicator during the h-adaptive procedures. A number of test cases are presented to assess the performance of the high-order h-adaptive rDG method. The hybrid reconstruction strategy combined with h-adaptive techniques presented in this work demonstrates promise for improving the level of accuracy and reducing the computational cost for numerical simulations of compressible inviscid flows.

A residual a posteriori error estimate for the Virtual Element Method

A residual-based a posteriori error estimate for the Poisson problem with discontinuous diffusivity coefficient is derived in the case of a virtual element discretization. The error is measured considering a suitable polynomial projection of the discrete solution to prove an equivalence between the defined error and a computable residual based error estimator that does not involve any term related to the virtual element stabilization. Numerical results display a very good behavior of the ratio between the error and the error estimator, resulting independent of the meshsize and element distortion.

A mixed higher order FEM for fully coupled compressible transversely isotropic finite hyperelasticity

A higher order mixed finite element method is presented for compressible transversely isotropic finite hyperelasticity. The independent variables of the three-field formulation are; displacement, fibre tension and fibre stretch. The formulation admits the description of a fully coupled stress response which evolves from an almost isotropic one into a hyper-anisotropic simply nearly inextensible response with increasing fibre tension, as for example observed in soft tissue biomechanics. Standard displacement approximation of order p in H1(Ω) is used while the auxiliary variables are approximated element wise by square integrable functions of order p−1 in L2(Ω). For finite extensibility the auxiliary variables are statically condensed out yielding a pure displacement based method. It is implemented in an hp-adaptive finite element code. A matching residual based error estimation capability is added and exercised. Numerical evidence indicating stability of approximation is supplied resolving a boundary layer caused by almost inextensible fibres. Coupled and uncoupled stress responses are compared. A generalised compressible transversely isotropic Holzapfel–Gasser–Ogden model is developed for the formulation that intentionally avoids the volumetric–isochoric split. Solutions obtained with the mixed method compare favourably to the corresponding ones obtained with pure displacement formulation. The latter fails in certain strongly anisotropic cases.

Mixed finite element methods for linear elasticity and the Stokes equations based on the Helmholtz decomposition

This paper introduces new mixed finite element methods (FEMs) of degree ≥1 for the equations of linear elasticity and the Stokes equations based on Helmholtz decompositions. These FEMs are robust with respect to the incompressible limit and also allow for mixed boundary conditions. Adaptive algorithms driven by efficient and reliable residual-based error estimators are introduced and proved to converge with optimal rate in the case of the Stokes equations with pure Dirichlet boundary.

An A Posteriori Analysis of $C^0$ Interior Penalty Methods for the Obstacle Problem of Clamped Kirchhoff Plates

We develop an a posteriori analysis of $C^0$ interior penalty methods for the displacement obstacle problem of clamped Kirchhoff plates. We show that a residual based error estimator originally designed for $C^0$ interior penalty methods for the boundary value problem of clamped Kirchhoff plates can also be used for the obstacle problem. We obtain reliability and efficiency estimates for the error estimator and introduce an adaptive algorithm based on this error estimator. Numerical results indicate that the performance of the adaptive algorithm is optimal for both quadratic and cubic $C^0$ interior penalty methods.

Axioms of Adaptivity with Separate Marking for Data Resolution

Mixed finite element methods with flux errors in $H({div})$-norms and div-least-squares finite element methods require a separate marking strategy in obligatory adaptive mesh-refining. The refinement indicator $\sigma^2(\mathcal{T},K)=\eta^2(\mathcal{T},K)+\mu^2(K)$ of a finite element domain $K$ in an admissible triangulation $\mathcal{T}$ consists of some residual-based error estimator $\eta(\mathcal{T},K)$ with some reduction property under local mesh-refining and some data approximation error $\mu(K)$. Separate marking means either Dorfler marking if $\mu^2(\mathcal{T})\leq\kappa\eta^2(\mathcal{T})$ or otherwise an optimal data approximation algorithm with controlled accuracy. The axioms are sufficient conditions on the estimators $\eta(\mathcal{T},K)$ and data approximation errors $\mu(K)$ for optimal asymptotic convergence rates. The enfolded set of axioms of this paper simplifies [C. Carstensen, M. Feischl, M. Page, and D. Praetorius, Comput. Math. Appl., 67 (2014), pp. 1195--1253] for collective m...

Residual based error estimate and quasi-interpolation on polygonal meshes for high order BEM-based FEM

Only a few numerical methods can treat boundary value problems on polygonal and polyhedral meshes. The BEM-based Finite Element Method is one of the new discretization strategies, which make use of and benefits from the flexibility of these general meshes that incorporate hanging nodes naturally. The article in hand addresses quasi-interpolation operators for the approximation space over polygonal meshes. To prove interpolation estimates the Poincaré constant is bounded uniformly for patches of star-shaped elements. These results give rise to the residual based error estimate for high order BEM-based FEM and its reliability as well as its efficiency are proven. Such a posteriori error estimates can be used to gauge the approximation quality and to implement adaptive FEM strategies. Numerical experiments show optimal rates of convergence for meshes with non-convex elements on uniformly as well as on adaptively refined meshes.

Multi-level hp-adaptivity and explicit error estimation

Recently, a multi-level hp-version of the finite element method (FEM) was proposed to ease the difficulties of treating hanging nodes, while providing full hp-approximation capabilities. In the original paper, the refinement procedure made use of a-priori knowledge of the solution. However, adaptive procedures can produce discretizations which are more effective than an intuitive choice of element sizes h and polynomial degree distributions p. This is particularly prominent when a-priori knowledge of the solution is only vague or unavailable. The present contribution demonstrates that multi-level hp-adaptive schemes can be efficiently driven by an explicit a-posteriori error estimator. To this end, we adopt the classical residual-based error estimator. The main insight here is that its extension to multi-level hp-FEM is possible by considering the refined-most overlay elements as integration domains. We demonstrate on several two- and three-dimensional examples that exponential convergence rates can be obtained.

Superconvergent patch recovery and a posteriori error estimation technique in adaptive isogeometric analysis

In this article, we address adaptive methods for isogeometric analysis based on local refinement guided by recovery based a posteriori error estimates.Isogeometric analysis was introduced a decade ago and an impressive progress has been made related to many aspects of numerical methods and advanced applications. However, related to adaptive mesh refinement guided by a posteriori error estimators, rather few attempts are pursued besides the use of classical residual based error estimators. In this article, we explore a feature common for Isogeometric analysis (IGA), namely the use of structured tensorial meshes that facilitates superconvergence behavior of the gradient in the Galerkin discretization. By utilizing the concept of structured mesh refinement using LR B-splines, our aim is to facilitate superconvergence behavior for locally refined meshes as well. Superconvergence behavior matches well with the use of recovery based a posteriori estimator in the Superconvergent Patch Recovery (SPR) procedure. However, to our knowledge so far, the SPR procedure has not been exploited in the IGA community.We start out by addressing the existence of derivative superconvergent points in the computed finite element solution based on B-splines and LR B-splines for an elliptic model problem (1D and 2D Poisson). Then, we present some recovery procedures for improving the derivatives (or gradient) of the isogeometric finite element solution where the SPR procedure will be the main focus. In particular, we show that our SPR procedure for the improvement of derivatives fulfills the desired consistency criteria. At the end, we develop a posteriori error estimator where the improved gradient obtained from the proposed recovery procedures is used.Numerical results are presented to illustrate the efficiency of using SPR procedure for the improvement of derivatives (or gradient) of computed solution in isogeometric analysis. Then the proposed a posteriori error estimator based adaptive refinement methodology is tested to solve smooth and non-smooth elliptic benchmark problems. The focus is put on whether optimal convergence rates are obtained in the computed solution or not, as well as the effectivity index of the proposed error estimators.

An a posteriori-driven adaptive Mixed High-Order method with application to electrostatics

In this work we propose an adaptive version of the recently introduced Mixed High-Order method and showcase its performance on a comprehensive set of academic and industrial problems in computational electromagnetism. The latter include, in particular, the numerical modeling of comb-drive and MEMS devices. Mesh adaptation is driven by newly derived, residual-based error estimators. The resulting method has several advantageous features: It supports fairly general meshes, it enables arbitrary approximation orders, and has a moderate computational cost thanks to hybridization and static condensation. The a posteriori-driven mesh refinement is shown to significantly enhance the performance on problems featuring singular solutions, allowing to fully exploit the high-order of approximation.

A five-field finite element formulation for nearly inextensible and nearly incompressible finite hyperelasticity

A novel Hu–Washizu type 5-field virtual work principle for nearly inextensible and almost incompressible finite hyperelasticity is developed and implemented. The formulation is implemented in an hp-adaptive code providing the proper flexible environment for finite elements with variable order and mixed interpolation. The novel implementation includes residual based error estimation and mesh adaptivity. In the fully constrained limit the formulation provides the constraint manifold setting of hyperelasticity with the simple internal kinematic constraints of inextensibility and incompressibility. A study using a semi-inverse analytical solution and h-refinements and p-enrichments corroborates the convergence characteristics of the 5-field implementation. A new closed form solution for pure torsion of a circular cylindrical tube with inextensible fibres is derived and used for verification. As typical applications are found in e.g. soft tissue biomechanics, passive pressurisation of an ellipsoidal geometry resembling the left ventricle of a rabbit heart is analysed to demonstrate the capability of modelling transversely isotropic materials.

Interpolation of Inverse Operators for Preconditioning Parameter-Dependent Equations

We propose a method for the construction of preconditioners of parameter-dependent matrices for the solution of large systems of parameter-dependent equations. The proposed method is an interpolation of the matrix inverse based on a projection of the identity matrix with respect to the Frobenius norm. Approximations of the Frobenius norm using random matrices are introduced in order to handle large matrices. The resulting statistical estimators of the Frobenius norm yield quasi-optimal projections that are controlled with high probability. Strategies for the adaptive selection of interpolation points are then proposed for different objectives in the context of projection-based model order reduction methods: the improvement of residual-based error estimators, the improvement of the projection on a given reduced approximation space, or the recycling of computations for sampling based model order reduction methods.

Some error estimates for the reproducing kernel Hilbert spaces method

In this paper we derive some effective error estimates for the reproducing kernel Hilbert space method applied to a general class of linear initial or boundary value problems. The first error estimate is computable and yields a worst case bound in the form of a percentage of the norm of the true solution which has not yet been discussed according to the knowledge of the authors. The second error estimate is a residual based error estimate, which is expressed in terms of the fill distance, so that convergence is studied for the fill distance tends to zero. This is a generalization and improvement of the existing error estimates. Some numerical results are presented to demonstrate the applicability of the estimates.

A discontinuous Galerkin method for optimal control problems governed by a system of convection–diffusion PDEs with nonlinear reaction terms

In this paper, we study the numerical solution of optimal control problems governed by a system of convection–diffusion PDEs with nonlinear reaction terms, arising from chemical processes. The symmetric interior penalty Galerkin (SIPG) method with upwinding for the convection term is used as a discretization method. We use a residual-based error estimator for the state and the adjoint variables. An adaptive mesh refinement indicated by a posteriori error estimates is applied. The arising saddle point system is solved using a suitable preconditioner. Numerical results are presented to illustrate the performance of the proposed error estimator.

Residual based error estimates for the space–time discontinuous Galerkin method applied to the compressible flows

We develop an adaptive numerical method for solution of the non-stationary compressible Navier–Stokes equations. This method is based on the space–time discontinuous Galerkin discretization, which employs high polynomial approximation degrees with respect to the space as well as to the time coordinates. We focus on the identification of the computational errors, following from the space and time discretizations and from the inexact solution of the arising nonlinear algebraic systems. We derive the residual-based error estimates approximating these errors. Then we propose an efficient algorithm which brings the algebraic, spatial and temporal errors under control. The computational performance of the proposed method is demonstrated by numerical experiments.

Numerical Approximation of Phase Field Based Shape and Topology Optimization for Fluids

We consider the problem of finding optimal shapes of fluid domains. The fluid obeys the Navier--Stokes equations. Inside a holdall container we use a phase field approach using diffuse interfaces to describe the domain of free flow. We formulate a corresponding optimization problem where flow outside the fluid domain is penalized. The resulting formulation of the shape optimization problem is shown to be well-posed, hence there exists a minimizer, and first order optimality conditions are derived. For the numerical realization we introduce a mass conserving gradient flow and obtain a Cahn--Hilliard type system, which is integrated numerically using the finite element method. An adaptive concept using reliable, residual based error estimation is exploited for the resolution of the spatial mesh. The overall concept is numerically investigated and comparison values are provided.

Numerical Investigations of an Error Bound for Reduced Basis Approximations of Noncoercice Variational Inequalities

We consider variational inequalities with different trial and test spaces and a possibly noncoercive bilinear form. Well-posedness has been shown under general conditions that are e.g. valid for the space-time formulation of parabolic variational inequalities. Fine discretizations for such problems resolve in large scale problems and thus in long computing times. To reduce the size of these problems, we use the Reduced Basis Method (RBM). Combining the RBM with the space-time formulation, a residual based error estimator has been derived in [Glas and Urban (2014)]. In this paper, we provide corresponding numerical results for a parametrized heat inequality model. Particularly, we perform two experiments concerning the error estimator. In the first one, we focus on rigor and efficiency of the error estimator depending on the specific method used for the basis generation and on the shape of the obstacle. In the second one, we show the quantitative reduction using the RBM in this setting.