Svetlana Matculevich

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.

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.

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.

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, Monika Wolfmayr

This work is aimed at the derivation of reliable and efficient a posteriori error estimates for convection-dominated diffusion problems motivated by a linear Fokker-Planck problem appearing in computational neuroscience. We obtain computable error bounds of the functional type for the static and time-dependent case and for different boundary conditions (mixed and pure Neumann boundary conditions). Finally, we present a set of various numerical examples including discussions on mesh adaptivity and space-time discretisation. The numerical results confirm the reliability and efficiency of the error estimates derived.

Svetlana Matculevich

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 efficient a posteriori error bound generation-based on solving an auxiliary problem with respect to an introduced vector-valued variable. 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 the routines for an auxiliary variable reconstruction which generate the majorant of an error can be executed several times faster than the routines for a primal unknown.

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.

Bärbel Holm, Svetlana Matculevich

This work is focused on the application of functional-type a posteriori error estimates and corresponding indicators to a class of time-dependent problems. We consider the algorithmic part of their derivation and implementation and also discuss the numerical properties of these bounds that comply with obtained numerical results. This paper examines two different methods of approximate solution reconstruction for evolutionary models, i.e., a time-marching technique and a space-time approach. The first part of the study presents an algorithm for global minimization of the majorant on each of discretization time-cylinders (time-slabs), the effectiveness of this algorithm is confirmed by extensive numerical tests. In the second part of the publication, the application of functional error estimates is discussed with respect to a space-time approach. It is followed by a set of extensive numerical tests that demonstrate the efficiency of proposed error control method.