Publications
REFEREED PUBLICATIONS IN JOURNALS
5.  Matculevich, S., and M. Wolfmayr On the a posteriori error analysis for linear FokkerPlanck models in convectiondominated diffusion problems, Applied Mathematics and Computation, 339, pp. 779804, 2018. Abstract This work is aimed at the derivation of reliable and efficient a posteriori error estimates for convectiondominated 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 timedependent 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 spacetime discretisation. The numerical results confirm the reliability and efficiency of the error estimates derived. / Download 
4.  Holm, B., and S. Matculevich Fully reliable error control for evolutionary problems, Comput. Math. Appl. (CAMWA), 75/4, pp. 13021329, 2018. Abstract This work is focused on the application of functionaltype a posteriori error estimates and corresponding indicators to a class of timedependent 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 timemarching 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 timecylinders (timeslabs), 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.
/ DownloadThe 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. 
3.  Matculevich, S. Journal of Computational and Applied Mathematics, Journal of Computational and Applied Mathematics, 344, pp. 394423, 2018. Abstract This work presents a numerical study of functional type a posteriori error estimates for IgA approximation schemes in the context of elliptic boundaryvalue 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, residualbased 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.

2.  Matculevich, S., P. Neitaanmäki, and S. Repin A posteriori error estimates for timedependent reactiondiffusion problems based on the PayneWeinberger inequality, Discrete Contin. Dyn. Syst. – Series A, 35/6, pp. 26592677, 2015. Abstract We consider evolutionary reactiondiffusion problems with mixed DirichletRobin boundary conditions. For this class of problems, we derive twosided 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. / Download 
1.  Matculevich, S., and S. Repin Computable estimates of the distance to the exact solution of the evolutionary reactiondiffusion equation, Appl. Math. Comput., 247, pp. 329347, 2014. Abstract We derive guaranteed bounds of distance to the exact solution of the evolutionary reactiondiffusion problem with mixed DirichletNeumann boundary condition. It is shown that twosided error estimates are directly computable and equivalent to the error. Numerical experiments confirm that estimates provide accurate twosided bounds of the overall error and generate efficient indicators of local error distribution. / Download 
REFEREED PUBLICATIONS IN JOURNALS
5.  Matculevich, S., and M. Wolfmayr On the a posteriori error analysis for linear FokkerPlanck models in convectiondominated diffusion problems, Applied Mathematics and Computation, 339, pp. 779804, 2018. Abstract This work is aimed at the derivation of reliable and efficient a posteriori error estimates for convectiondominated 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 timedependent 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 spacetime discretisation. The numerical results confirm the reliability and efficiency of the error estimates derived. / Download 
4.  Holm, B., and S. Matculevich Fully reliable error control for evolutionary problems, Comput. Math. Appl. (CAMWA), 75/4, pp. 13021329, 2018. Abstract This work is focused on the application of functionaltype a posteriori error estimates and corresponding indicators to a class of timedependent 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 timemarching 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 timecylinders (timeslabs), 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.
/ DownloadThe 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. 
3.  Matculevich, S. Journal of Computational and Applied Mathematics, Journal of Computational and Applied Mathematics, 344, pp. 394423, 2018. Abstract This work presents a numerical study of functional type a posteriori error estimates for IgA approximation schemes in the context of elliptic boundaryvalue 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, residualbased 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.

2.  Matculevich, S., P. Neitaanmäki, and S. Repin A posteriori error estimates for timedependent reactiondiffusion problems based on the PayneWeinberger inequality, Discrete Contin. Dyn. Syst. – Series A, 35/6, pp. 26592677, 2015. Abstract We consider evolutionary reactiondiffusion problems with mixed DirichletRobin boundary conditions. For this class of problems, we derive twosided 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. / Download 
1.  Matculevich, S., and S. Repin Computable estimates of the distance to the exact solution of the evolutionary reactiondiffusion equation, Appl. Math. Comput., 247, pp. 329347, 2014. Abstract We derive guaranteed bounds of distance to the exact solution of the evolutionary reactiondiffusion problem with mixed DirichletNeumann boundary condition. It is shown that twosided error estimates are directly computable and equivalent to the error. Numerical experiments confirm that estimates provide accurate twosided bounds of the overall error and generate efficient indicators of local error distribution. / Download 