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Chapter
A Posteriori Error Estimates for Domain Decomposition Methods
Nowadays, a posteriori error control methods have formed a new important part of the numerical analysis. Their purpose is to obtain computable error estimates in various norms and error indicators that show di...
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Chapter
Artificial Intelligence and Computational Science
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 mat...
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Chapter
Generation of Error Indicators for Partial Differential Equations by Machine Learning Methods
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, erro...
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Article
Open AccessFunctional a posteriori error estimates for boundary element methods
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...
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Chapter
Poincaré Type Inequalities for Vector Functions with Zero Mean Normal Traces on the Boundary and Applications to Interpolation Methods
We consider inequalities of the Poincaré–Steklov type for subspaces of \(H^1\) -functions defined in a bounded domain ...
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Chapter and Conference Paper
Verifications of Primal Energy Identities for Variational Problems with Obstacles
We discuss error identities for two classes of free boundary problems generated by obstacles. The identities suggest true forms of the respective error measures which consist of two parts: standard energy norm...
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Chapter and Conference Paper
Functional Type Error Control for Stabilised Space-Time IgA Approximations to Parabolic Problems
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 approxima...
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Article
Guaranteed and computable bounds of the limit load for variational problems with linear growth energy functionals
The paper is concerned with guaranteed and computable bounds of the limit (or safety) load, which is one of the most important quantitative characteristics of mathematical models associated with linear growth ...
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Book
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Chapter
Error Estimates of Uzawa Iteration Method for a Class of Bingham Fluids
The paper is concerned with fully guaranteed and computable bounds of errors generated by Uzawa type methods for variational problems in the theory of visco-plastic fluids. The respective estimates have two fo...
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Chapter and Conference Paper
Estimates of Constants in Boundary-Mean Trace Inequalities and Applications to Error Analysis
We discuss Poincaré type inequalities for functions with zero mean values on the whole boundary of a Lipschitz domain or on a measurable part of the boundary. For some basic domains (rectangles, quadrilaterals...
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Book
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Chapter
Errors Arising in Computer Simulation Methods
The goal of this introductory chapter is to discuss in general terms different classes of errors arising in computer simulation methods and to direct the reader to the chapters and sections of the book where t...
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Chapter
Two-Sided Guaranteed Estimates of the Cost Functional for Optimal Control Problems with Elliptic State Equations
In the paper, we discuss error estimation methods for optimal control problems with distributed control functions entering the right-hand side of the corresponding elliptic state equations. Our analysis is bas...
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Chapter
Indicators of Errors for Approximate Solutions of Differential Equations
Error indicators play an important role in mesh-adaptive numerical algorithms, which currently dominate in mathematical and numerical modeling of various models in physics, chemistry, biology, economics, and o...
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Chapter
Guaranteed Error Bounds II
In Chap. 3, we discussed the main ideas of fully reliable error control methods and the corresponding numerical algorithms with the paradigm of simple elliptic type prob...
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Chapter
Overview of Other Results and Open Problems
This chapter presents an overview of results related to error control methods, which were not considered in previous chapters. In the first part, we discuss possible extensions of the theory exposed in Chaps. ...
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Chapter
A Unified Approach to Measuring Accuracy of Error Indicators
In this paper, we present a unified approach to error indication for elliptic boundary value problems. We introduce two different definitions of the accuracy (weak and strong) and show that various indicators ...
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Chapter
Guaranteed Error Bounds I
In this chapter, we discuss foundations of new error control methods developed during the last 10–12 years. First, we consider the simplest boundary value problems generated by ordinary differential equations ...
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Chapter
Errors Generated by Uncertain Data
In this chapter, we study effects caused by incompletely known data. In practice, the data are never known exactly, therefore the results generated by a mathematical model also have a limited accuracy. Then, t...
