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The inf–sup (or LBB) condition plays a crucial role in analysis of viscous flow problems and other problems related to incompressible media. In this paper, we deduce localized forms of this condition that contain a collection of local constants associated with subdomains instead of one global constant for the whole domain. Localized forms of the LBB inequality imply estimates of the distance to the set of divergence free fields. We use them and deduce fully computable bounds of the distance between approximate and exact solutions of boundary value problems arising in the theory of viscous incompressible fluids. The estimates are valid for approximations, which satisfy the incompressibility condition only in a very weak (integral) form. Another important question considered in the paper is how to select proper measures that should be used in error analysis. We show that such a measure is dictated by the respective error identity and discuss properties of the measure for the Stokes, Oseen, and Navier–Stokes problems.
We consider inequalities of the Poincaré–Steklov type for subspaces of H1 -functions defined in a bounded domain Ω∈Rd with Lipschitz boundary ∂Ω . For scalar valued functions, the subspaces are defined by zero mean condition on ∂Ω or on a part of ∂Ω having positive d−1 measure. For vector valued functions, zero mean conditions are applied to normal components on plane faces of ∂Ω (or to averaged normal components on curvilinear faces). We find explicit and simply computable bounds of constants in the respective Poincaré type inequalities for domains typically used in finite element methods (triangles, quadrilaterals, tetrahedrons, prisms, pyramids, and domains composed of them). The second part of the paper discusses applications of the estimates to interpolation of scalar and vector valued functions on macrocells and on meshes with non-overlapping and overlapping cells.
A posteriori error estimation methods are usually developed in the context of upper and lower bounds of errors. In this paper, we are concerned with a posteriori analysis in terms of identities, i.e., we deduce error relations, which holds as equalities. We discuss a general form of error identities for a wide class of convex variational problems. The left hand sides of these identities can be considered as certain measures of errors (expressed in terms of primal/dual solutions and respective approximations) while the right hand sides contain only known approximations. Finally, we consider several examples and show that in some simple cases these identities lead to generalized forms of the Prager-Synge and Mikhlin’s error relations. Also, we discuss particular cases related to power growth functionals and to the generalized Stokes problem.
We consider an iteration method for solving an elliptic type boundary value problem Au=f, where a positive definite operator A is generated by a quasi-periodic structure with rapidly changing coefficients (a typical period is characterized by a small parameter ϵ). The method is based on using a simpler operator A0 (inversion of A0 is much simpler than inversion of A), which can be viewed as a preconditioner for A. We prove contraction of the iteration method and establish explicit estimates of the contraction factor q. Certainly the value of q depends on the difference between A and A0. For typical quasi-periodic structures, we establish simple relations that suggest an optimal A0 (in a selected set of “simple” structures) and compute the corresponding contraction factor. Further, this allows us to deduce fully computable two-sided a posteriori estimates able to control numerical solutions on any iteration. The method is especially efficient if the coefficients of A admit low-rank representations and if algebraic operations are performed in tensor structured formats. Under moderate assumptions the storage and solution complexity of our approach depends only weakly (merely linear-logarithmically) on the frequency parameter 1/ϵ.
Abstract: We derive computable bounds of deviations from the exact solution of the stationary Oseen prob- lem. They are applied to approximations generated by the Uzawa iteration method. Also, we derive an ad- vanced form of the estimate, which takes into account approximation errors arising due to discretization of the boundary value problem, generated by the main step of the Uzawa method. Numerical tests con rm our theoretical results and show practical applicability of the estimates.
In this note, we discuss the interaction between two ways of scientific analysis. The first (classical) way is known as Mathematical Modeling (MM). It is based on a model created by humans and presented in mathematical terms. Scientific Computing (SC) is an important tool of MM developed to quantitatively analyze the model. Artificial Intelligence (AI) forms a new way of scientific analysis. AI systems arise as a result of a different process. Here, we take a sequence of correct input–output data, perform Machine Learning (ML), and get a model (hidden in a network). In this process, computational methods are used to create a network type model. We briefly discuss special methods used for this purpose (such as evolutionary algorithms), give a concise overview of results related to applications of AI in computer simulation of real-life problems, and discuss several open problems.
We consider elliptic variational inequalities generated by obstacle type problems with thin obstacles. For this class of problems, we deduce estimates of the distance (measured in terms of the natural energy norm) between the exact solution and any function that satisfies the boundary condition and is admissible with respect to the obstacle condition (i.e., they are valid for any approximation regardless of the method by which it was found). Computation of the estimates does not require knowledge of the exact solution and uses only the problem data and an approximation. The estimates provide guaranteed upper bounds of the error (error majorants) and vanish if and only if the approximation coincides with the exact solution. In the last section, the efficiency of error majorants is confirmed by an example, where the exact solution is known.
The paper is concerned with reliable space-time IgA schemes for parabolic initial-boundary value problems. We deduce a posteriori error estimates and investigate their applicability to space-time IgA approximations. Since the derivation is based on purely functional arguments, the estimates do not contain mesh dependent constants and are valid for any approximation from the admissible (energy) class. In particular, they imply estimates for discrete norms associated with stabilised space-time IgA approximations. Finally, we illustrate the reliability and efficiency of presented error estimates for the approximate solutions recovered with IgA techniques on a model example.
We consider elliptic problems with complicated, discontinuous diffusion tensor A0. One of the standard approaches to numerically treat such problems is to simplify the coefficient by some approximation, say Aε, and to use standard finite elements. In [19] a combined modelling-discretization strategy has been proposed which estimates the discretization and modelling errors by a posteriori estimates of functional type. This strategy allows to balance these two errors in a problem adapted way. However, the estimate of the modelling error is derived under the assumption that the difference A0 − Aε is bounded in the L∞-norm, which requires that the approximation of the coefficient matches the discontinuities of the original coefficient. Therefore this theory is not appropriate for applications with discontinuous coefficients along complicated, curved interfaces. Based on bounds for A0 − Aε in an L q -norm with q < ∞ we generalize the combined modelling-discretization strategy to a larger class of coefficients.
The paper is concerned with a coupled problem describing piesoelectric effects in an elastic body. For this problem, we deduce majorants of the distance between the exact solution and any approximation in the respective energy class of functions satisfying the boundary conditions. The majorants are fully computable and does not contain mesh dependent constants. They vanish if and only if an approximate solution coincides with the exact one and provide a realistic measure of the accuracy in terms of the natural energy norm associated with the coupled problem studied.
In this paper, we derive computable a posteriori error bounds for approximations computed by the Uzawa algorithm for the generalized Stokes problem. We show that for each Uzawa iteration both the velocity error and the pressure error are bounded from above by a constant multiplied by the L2-norm of the divergence of the velocity. The derivation of the estimates essentially uses a posteriori estimates of the functional type for the Stokes problem.
The aim of this paper is to introduce an enhanced incremental procedure that can be used for the numerical evaluation and reliable estimation of the limit load. A conventional incremental method of limit analysis is based on parametrization of the respective variational formulation by the loading parameter ζ ∈ (0, ζlim), where ζlim is generally unknown. The enhanced incremental procedure is operated in terms of an inverse mapping ψ : α 7→ ζ where the parameter α belongs to (0, +∞) and its physical meaning is work of applied forces at the equilibrium state. The function ψ is continuous, nondecreasing and its values tend to ζlim as α → +∞. Reduction of the problem to a finite element subspace associated with a mesh Th generates the discrete limit parameter ζlim,h and the discrete counterpart ψh to the function ψ. We prove pointwise convergence ψh → ψ and specify a class of yield functions for which ζlim,h → ζlim. These convergence results enable to find reliable lower and upper bounds of ζlim. Numerical tests confirm computational efficiency of the suggested method.
The paper is concerned with space–time IgA approximations to parabolic initial–boundary value problems. We deduce guaranteed and fully computable error bounds adapted to special features of such type of approximations and investigate their efficiency. The derivation of error estimates is based on the analysis of the corresponding integral identity and exploits purely functional arguments in the maximal parabolic regularity setting. The estimates are valid for any approximation from the admissible (energy) class and do not contain mesh-dependent constants. They provide computable and fully guaranteed error bounds for the norms arising in stabilised space–time approximations. Furthermore, a posterior error estimates yield efficient error indicators enhancing the performance of adaptive solvers and generate very successful mesh refinement procedures. Theoretical results are verified with a series of numerical examples, in which approximate solutions and the corresponding fluxes are recovered by IgA techniques. The numerical results confirm the high efficiency of the method in the context of the two main goals of a posteriori error analysis: estimation of global errors and mesh adaptation.
The paper is devoted to a family of specific inf–sup conditions generated by tensor-valued functions on convex cones. First, we discuss the validity of such conditions and estimate the value of the respective constant. Then, the results are used to derive estimates of the distance to dual cones, which are required in the analysis of limit loads of perfectly plastic structures. The equivalence between the static and kinematic approaches to limit analysis is proven and computable majorants of the limit load are derived. Particular interest is paid to the Drucker–Prager yield criterion. The last section exposes a collection of numerical examples including basic geotechnical stability problems. The majorants of the limit load are computed and expected failure mechanisms of structures are visualized using local mesh adaptivity.
The paper is concerned with an elliptic variational inequality associated with a free boundary obstacle problem for the biharmonic operator. We study the bounds of the difference between the exact solution (minimizer) of the corresponding variational problem and any function (approximation) from the energy class satisfying the prescribed boundary conditions and the restrictions stipulated by the obstacle. Using the general theory developed for a wide class of convex variational problems we deduce the error identity. One part of this identity characterizes the deviation of the function (approximation) from the exact solution, whereas the other is a fully computed value (it depends only on the data of the problem and known functions). In real life computations, this identity can be used to control the accuracy of approximate solutions. The measure of deviation from the exact solution used in the error identity contains terms of different nature. Two of them are the norms of the difference between the exact solutions (of the direct and dual variational problems) and corresponding approximations. Two others are not representable as norms. These are nonlinear measures vanishing if the coincidence set defined by means of an approximate solution satisfies certain conditions (for example, coincides with the exact coincidence set). The error identity is true for any admissible (conforming) approximations of the direct variable, but it imposes some restrictions on the dual variable. We show that these restrictions can be removed, but in this case the identity is replaced by an inequality. For any approximations of the direct and dual variational problems, the latter gives an explicitly computable majorant of the deviation from the exact solution. Several examples illustrating the established identities and inequalities are presented.
The contribution is devoted to computations of the limit load for a perfectly plastic model with the von Mises yield criterion. The limit factor of a prescribed load is defined by a specific variational problem, the so-called limit analysis problem. This problem is solved in terms of deformation fields by a penalization, the finite element and the semismooth Newton methods. From the numerical solution, we derive a guaranteed upper bound of the limit factor. To achieve more accurate results, a local mesh adaptivity is used.
Computer simulation methods for models based on partial differential equations usually apply adaptive strategies that generate sequences of approximations for consequently refined meshes. In this process, error indicators play a crucial role because a new (refined) mesh is created by analysis of an approximate solution computed for the previous (coarser) mesh. Different error indicators exploit various analytical and heuristic arguments. The main goal of this paper is to show that effective indicators of approximation errors can be created by machine learning methods and presented by relatively simple networks. We use the “supervised learning” conception where sequences of teaching examples are constructed due to earlier developed tools of a posteriori error analysis known as “functional type error majorants”. Insensitivity to specific features of approximations is an important property of error majorants, which allows us to generate arbitrarily long series of diverse training examples without restrictions on the type of approximate solutions. These new (network) error indicators are compared with known indicators. The results show that after a proper machine learning procedure, we obtain a network with the same (or even better) quality of error indication level as the most efficient indicators used in classical computer simulation methods. The final trained network is approximately as effective as the gradient averaging error indicator, but has an important advantage because it is valid for a much wider set of approximate solutions.
In this paper, we derive a posteriori bounds of the di erence between the exact solution of an elliptic boundary value problem with periodic coe cients and an abridged model, which follows from the homogenization theory. The di erence is measured in terms of the energy norm of the basic problem and also in the combined primal–dual norm. Using the technique of functional type a posteriori error estimates, we obtain two-sided bounds of the modeling error, which depends only on known data and the solution of the homogenized problem. It is proved that the majorant with properly chosen arguments possesses the same convergence rate, which was established for the true error. Numerical tests con rm the effi ciency of the estimates.
We present a new version of the Picard-Lindelof method for ordinary dif- ¨ ferential equations (ODEs) supplied with guaranteed and explicitly computable upper bounds of an approximation error. The upper bounds are based on the Ostrowski estimates and the Banach fixed point theorem for contractive operators. The estimates derived in the paper take into account interpolation and integration errors and, therefore, provide objective information on the accuracy of computed approximations.
We consider a class of nonlinear elliptic problems associated with models in biophysics, which are described by the Poisson–Boltzmann equation (PBE). We prove mathematical correctness of the problem, study a suitable class of approximations, and deduce guaranteed and fully computable bounds of approximation errors. The latter goal is achieved by means of the approach suggested in [] for convex variational problems. Moreover, we establish the error identity, which defines the error measure natural for the considered class of problems and show that it yields computable majorants and minorants of the global error as well as indicators of local errors that provide efficient adaptation of meshes. Theoretical results are confirmed by a collection of numerical tests that includes problems on 2D and 3D Lipschitz domains.
This article is devoted to the a posteriori error analysis of multiharmonic finite element approximations to distributed optimal control problems with time-periodic state equations of parabolic type. We derive a posteriori estimates of the functional type, which are easily computable and provide guaranteed upper bounds for the state and co-state errors as well as for the cost functional. These theoretical results are confirmed by several numerical tests that show high efficiency of the a posteriori error bounds.
In this paper we present a new error indicator for approximate solutions of elliptic problems. We discuss error indication with the paradigm of the diffusion problem, however the techniques are easily adaptable to more complicated elliptic problems, for example to linear elasticity, viscous flow models and electromagnetic models. The proposed indicator does not contain mesh dependent constants and it admits parallelization.
The paper is concerned with guaranteed a posteriori error estimates for a class of evolutionary problems related to poroelastic media governed by the quasi-static linear Biot equations. The system is decoupled by employing the fixed-stress split scheme, which leads to an iteratively solved semi-discrete system. The error bounds are derived by combining a posteriori estimates for contractive mappings with functional type error control for elliptic partial differential equations. The estimates are applicable to any approximation in the admissible functional space and are independent of the discretization method. They are fully computable, do not contain mesh-dependent constants, and provide reliable global estimates of the error measured in the energy norm. Moreover, they suggest efficient error indicators for the distribution of local errors and can be used in adaptive procedures.
Functional error estimates are well-established tools for a posteriori error estimation and related adaptive mesh-refinement for the finite element method (FEM). The present work proposes a first functional error estimate for the boundary element method (BEM). One key feature is that the derived error estimates are independent of the BEM discretization and provide guaranteed lower and upper bounds for the unknown error. In particular, our analysis covers Galerkin BEM and the collocation method, what makes the approach of particular interest for scientific computations and engineering applications. Numerical experiments for the Laplace problem confirm the theoretical results.
In this paper, we discuss a posteriori estimates for the Maxwell type boundary-value problem. The estimates are derived by transformations of integral identities that define the generalized solution and are valid for any conforming approximation of the exact solution. It is proved analytically and confirmed numerically that the estimates indeed provide a computable and guaranteed bound of approximation errors. Also, it is shown that the estimates imply robust error indicators that represent the distribution of local (inter-element) errors measured in terms of different norms.