Sergey Repin

Ulrich Langer, Sergey Repin, Monika Wolfmayr

This paper 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 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.

Jaroslav Haslinger, Sergey Repin, Stanislav Sysala

The aim of this paper is to introduce a new incremental procedure that can be used for numerical evaluation of the limit load. Existing incremental type methods are based on parametrization of the energy by the loading parameter $ζ\in[0,ζ_{lim})$, where $ζ_{lim}$ is generally unknown. In the new method, the incremental procedure is operated in terms of an inverse mapping and the respective parameter $α$ is changing in the interval $(0,+\infty)$. Theoretically, in each step of this algorithm, we obtain a guaranteed lower bound of $ζ_{lim}$. Reduction of the problem to a finite element subspace associated with a mesh $\mathcal T_h$ generates computable bound $ζ_{lim,h}$. Under certain assumptions, we prove that $ζ_{lim,h}$ tends to $ζ_{lim}$ as $h\rightarrow0_+$. Numerical tests confirm practical efficiency of the suggested method.

Dirk Pauly, Sergey Repin

This paper is concerned with the derivation of computable and guaranteed upper bounds of the difference between the exact and the approximate solution of an exterior domain boundary value problem for a linear elliptic equation. Our analysis is based upon purely functional argumentation and does not attract specific properties of an approximation method. Therefore, the estimates derived in the paper at hand are applicable to any approximate solution that belongs to the corresponding energy space. Such estimates (also called error majorants of the functional type) have been derived earlier for problems in bounded domains.

Svetlana Matculevich, Sergey Repin

We derive guaranteed bounds of distance to the exact solution of the evolutionary reaction-diffusion problem with mixed Dirichlet-Neumann boundary condition. It is shown that two-sided error estimates are directly computable and equivalent to the error. Numerical experiments confirm that estimates provide accurate two-sided bounds of the overall error and generate efficient indicators of local error distribution.

Dirk Pauly, Sergey Repin

This paper is concerned with the derivation of computable and guaranteed upper and lower bounds of the difference between the exact and the approximate solution of a boundary value problem for static Maxwell equations. Our analysis is based upon purely functional argumentation and does not attract specific properties of an approximation method. Therefore, the estimates derived in the paper at hand are applicable to any approximate solution that belongs to the corresponding energy space. Such estimates (also called error majorants of the functional type) have been derived earlier for elliptic problems.

Ulrich Langer, Svetlana Matculevich, Sergey Repin

The paper is concerned with locally stabilized space-time IgA approximations to initial boundary value problems of the parabolic type. Originally, similar schemes (but weighted with a global mesh parameter) was presented and studied by U. Langer, M. Neumueller, and S. Moore (2016). The current work devises a localised version of this scheme and establishes coercivity, boundedness, and consistency of the corresponding bilinear form. Using these fundamental properties together with the corresponding approximation error estimates for B-splines, we show that the space-time IgA solutions generated by the new scheme satisfy asymptotically optimal a priori discretization error estimates. The adaptive mesh refinement algorithm proposed in the paper is based on a posteriori error estimates of the functional type that has been rigorously studied in earlier works by S. Repin (2002) and U. Langer, S. Matculevich, and S. Repin (2017). Numerical results presented in the second part of the paper confirm the improved convergence of global approximation errors. Moreover, these results also confirm the local efficiency of the error indicators produced by the error majorants.

Ulrich Langer, Svetlana Matculevich, Sergey Repin

This work is concerned with a posteriori error estimates of the functional type for approximations constructed by space-time IgA scheme presented in paper by Langer, Neumueller, and Moore (2016). We consider approxima- tions in the corresponding IgA spaces based on elliptic and bounded bilinear form (associated with the spatial part). It is proved that the approximations satisfy classic a priori error estimates. Also, we deduce a posteriori error estimates for a stabilized weak formulation of the considered parabolic initial boundary value problem (I-BVP). They are derived by a general functional method and do not contain mesh dependent constants. The estimates are valid for a wide class of approximations. In particular, they imply estimates for the discrete norm of IgA approximations. Moreover, we introduce different forms of a posteriori error estimates (error majorants) and establish equivalence of majorants and energy error norm. This property justifies efficiency and reliability of a posteriori error estimates. Another important property of the estimates is their flexibility with respect to a certain amount of free parameters. Using these parameters we can obtain estimates for different error norms and minimize the respective majorant in order to find the best possible bound of the error.

Johannes Kraus, Svetoslav Nakov, Sergey Repin

In this paper we have derived explicitly computable bounds on the error in energy norms for the fully nonlinear Poisson-Boltzmann equation. Together with the computable bounds, we have also obtained efficient error indicators which can serve as a basis for a reliable adaptive finite element algorithm.

Johannes Kraus, Svetoslav Nakov, Sergey Repin

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 [S. Repin, A posteriori error estimation for variational problems with uniformly convex functionals. Math. Comp., 69:481-500, 2000] 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.

Svetlana Matculevich, Sergey Repin

The paper is concerned with sharp estimates of constants in Poincare type inequalities for functions having zero mean value on the boundary of a Lipschitz domain or on a measurable part of it. These estimates are useful for various numerical methods, in particular, for a posteriori error estimation methods for partial differential equations (PDEs). Therefore, we are mainly focused on domains typical for numerical analysis (simplexes in 2d and 3d) and suggest easily computable relations that provide sharp bounds of the respective constants. Also, we investigate numerically the behavior of the constants in the classical Poincare inequalities and compare these results with known analytical estimates. In the last section, the estimates are used in order to obtain new a posteriori estimates for an elliptic boundary value problem.

Dirk Pauly, Sergey Repin, Tuomo Rossi

In this paper, we consider an initial boundary value problem for Maxwell's equations. For this hyperbolic type problem, we derive guaranteed and computable upper bounds for the difference between the exact solution and any pair of vector fields in the space-time cylinder that belongs to the corresponding admissible energy class. For this purpose, we use a method suggested earlier for the wave equation.

Dirk Pauly, Sergey Repin

This paper is concerned with the analysis of the inf-sup condition arising in the stationary Stokes problem in exterior domains. We deduce values of the constant in the stability lemma, which yields fully computable estimates of the distance to the set of divergence free fields defined in exterior domains. Using these estimates we obtain computable majorants of the difference between the exact solution of the Stokes problem in exterior domains and any approximation from the admissible (energy) class of functions satisfying the Dirichlet boundary condition exactly.

Svetlana Matculevich, Sergey Repin

The goal of the paper is to derive two-sided bounds of the distance between the exact solution of the evolutionary reaction-diffusion problem with mixed Dirichlet--Robin boundary conditions and any function in the admissible energy space. The derivation is based upon transformation of the integral identity, which defines the generalized solution, and exploits classical Poincare inequalities and Poincare type inequalities for functions with zero mean boundary traces. The corresponding constants are estimated due to Payne and Weinberger, 1960, and Nazarov and Repin, 2013. To handle problems with complex domains and mixed boundary conditions, domain decomposition is used. The corresponding bounds of the distance to the exact solution, contain only constants in local Poincare type inequalities associated with subdomains. Moreover, it is proved that the bounds are equivalent to the energy norm of the error.

Svetlana Matculevich, Sergey Repin

The goal of the paper is to derive two-sided bounds of the distance between the exact solution of the evolutionary reaction-diffusion problem with mixed Dirichlet--Robin boundary conditions and any function in the admissible energy space.

Ulrich Langer, Sergey Repin, Monika Wolfmayr

The paper is concerned with parabolic time-periodic boundary value problems which are of theoretical interest and arise in different practical applications. The multiharmonic finite element method is well adapted to this class of parabolic problems. We study properties of multiharmonic approximations and derive guaranteed and fully computable bounds of approximation errors. For this purpose, we use the functional a posteriori error estimation techniques earlier introduced by S. Repin. Numerical tests confirm the efficiency of the a posteriori error bounds derived.