REFEREED PUBLICATIONS IN JOURNALS
Kyas, S., U. Langer, and S. Repin Guaranteed error bounds and local indicators for adaptive solvers using stabilized space-time IgA approximations to parabolic problems Computers and Mathematics with Applications, 78/8, pp. 2641-2671, 2019. [Download PDF] [View Abstract]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 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. Theoretical results are verified on a series of numerical examples, in which approximate solutions and auxiliary fluxes are recovered by IgA techniques. The mesh refinement algorithm is governed by local error indicators that naturally follow from the global error majorants. The numerical results confirm 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.
Matculevich, S., and M. Wolfmayr On the a posteriori error analysis for linear Fokker-Planck models in convection-dominated diffusion problems Applied Mathematics and Computation, 339, pp. 779-804, 2018. [Download PDF] [View Abstract]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 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.
Holm, B., and S. Matculevich Fully reliable error control for evolutionary problems Comput. Math. Appl. (CAMWA), 75/4, pp. 1302-1329, 2018. [Download PDF] [View Abstract]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 solution approximation 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 minimisation of the majorant on each of discretisation time-cylinders (time-slabs), the effectiveness of this approach to error estimation 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 demonstrates the efficiency of proposed error control method. The numerical results obtained in this paper rely on the implementation carried out using open source software, which allows formulating the problem in a weak setting. To work in a variational framework appears to be very natural for functional error estimates due to their derivation method. The search for the optimal parameters for the majorant is done by a global functional minimisation, which to the authors’ knowledge is the first work using this technique in an evolutionary framework.
Matculevich, S. Functional approach to the error control in adaptive IgA schemes for elliptic boundary value problems Journal of Computational and Applied Mathematics, 344, pp. 394-423, 2018. [Download PDF] [View Abstract]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.
Matculevich, S., and S. Repin Explicit constants in Poincaré-type inequalities for simplicial domains and application to a posteriori estimates Comput. Methods Appl. Math. (CMAM), 16/2, pp. 277-298, 2016. [Download PDF] [View Abstract]The paper is concerned with sharp estimates of constants in the classical Poincaré inequalities and Poincaré-type inequalities for functions with zero mean values in a simplicial domain or on a part of the boundary. These estimates are important for quantitative analysis of problems generated by differential equations where numerical approximations are typically constructed with the help of simplicial meshes. We suggest easily computable relations that provide sharp bounds of the respective constants and compare these results with analytical estimates (if such estimates are known). In the last section, we discuss possible applications and derive a computable majorant of the difference between the exact solution of a boundary value problem and an arbitrary finite dimensional approximation defined on a simplicial mesh.
Matculevich, S., P. Neitaanmäki, and S. Repin A posteriori error estimates for time-dependent reaction-diffusion problems based on the Payne-Weinberger inequality Discrete Contin. Dyn. Syst. - Series A, 35/6, pp. 2659-2677, 2015. [Download PDF] [View Abstract]We consider evolutionary reaction-diffusion problems with mixed Dirichlet-Robin boundary conditions. For this class of problems, we derive two-sided estimates of the distance between any function in the admissible energy space and the exact solution of the problem. The estimates (majorants and minorants) are explicitly computable and do not contain unknown functions or constants. Moreover, it is proved that the estimates are equivalent to the energy norm of the deviation from the exact solution.
Matculevich, S., and S. Repin Computable estimates of the distance to the exact solution of the evolutionary reaction-diffusion equation Appl. Math. Comput., 247, pp. 329-347, 2014. [Download PDF] [View Abstract]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.
[back to Top of Page]
Kyas, S., S. Repin, and U. Langer Space-Time Isogeometric Analysis of Parabolic Diffusion Problems in Moving Spatial Domains , Proceedings in Applied Mathematics and Mechanics, 2019. [View Abstract]This paper is devoted to locally stabilized space-time isogeometric (IgA) schemes for parabolic diffusion problems in moving spatial domains. It generalizes the results of our preceding works for problems in fixed spatial domains. We present functional a posteriori error estimates and study adaptive numerical procedures based on them and on truncated hierarchical B-splines.
Kyas, S., U. Langer, and S. Repin Adaptive space-time isogeometric analysis for parabolic evolution problems , Space-Time Methods: Applications to Partial Differential Equations, Radon Series on Computational and Applied Mathematics, pp. 155-199, 2019. [View Abstract]The paper proposes new locally stabilized space-time Isogeometric Analysis approximations to initial boundary value problems of the parabolic type. Previously, similar schemes (but weighted with a global mesh parameter) has been presented and studied by U. Langer, M. Neumüller, and S. Moore (2016). The current work devises a localized version of this scheme, which is suited for adaptive mesh refinement. We establish coercivity, boundedness, and consistency of the corresponding bilinear form. Using these fundamental properties together with standard approximation error estimates for B-splines and NURBS, we show that the space-time Isogeometric Analysis solutions generated by the new scheme satisfy asymptotically optimal a priori discretization error estimates. Error indicators used for mesh refinement are based on a posteriori error estimates of the functional type that has been introduced by S. Repin (2002) and later rigorously studied in the context of Isogeometric Analysis by U. Langer, S. Matculevich, and S. Repin (2017). Numerical results discussed in the paper illustrate an improved convergence of global approximation errors and respective error majorants. They also confirm the local efficiency of the error indicators produced by the error majorants.
Matculevich, S., U. Langer, and S. Repin Functional Type Error Control for Stabilized Space-Time IgA Approximations to Parabolic Problems , Lecture Notes in Computer Science, 10665 LNCS, pp. 55-65, 2018. [Download PDF] [View Abstract]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.
Matculevich, S., P. Neitaanmäki, and S. Repin Guaranteed error bounds for a class of Picard-Lindelöf iteration methods , Numerical Methods for Diff. Eq., Opt., and Tech. Problems, Comput. Methods Appl. Sci., 27, pp. 151-168, 2013. [Download PDF] [View Abstract]We present a new version of the Picard-Lindelöf method for ordinary differential 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.