Journal ArticleDOI
A posteriori error estimates for finite element discretizations of the heat equation
TLDR
For these discretizations of the heat equation by A-stable θ-schemes in time and conforming finite elements in space, residual a posteriori error indicators are derived that yield upper bounds on the error and lower bounds that are global in space and time and local in time.Abstract:
We consider discretizations of the heat equation by A-stable θ-schemes in time and conforming finite elements in space. For these discretizations we derive residual a posteriori error indicators. The indicators yield upper bounds on the error which are global in space and time and yield lower bounds that are global in space and local in time. The ratio between upper and lower bounds is uniformly bounded in time and does not depend on any step-size in space or time. Moreover, there is no restriction on the relation between the step-sizes in space and time.read more
Citations
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Journal ArticleDOI
Robust A Posteriori Error Estimates for Stationary Convection-Diffusion Equations
TL;DR: All estimators yield global upper and lower bounds for the error measured in a norm that incorporates the standard energy norm and a dual norm of the convective derivative, and also hold for the limit case of vanishing reaction.
Journal ArticleDOI
A posteriori error estimates for the Crank–Nicolson method for parabolic equations
TL;DR: The resulting estimators are shown to be of optimal order by deriving upper and lower bounds for them depending only on the discretization parameters and the data of the problem.
Journal ArticleDOI
Guaranteed and robust a posteriori error estimates and balancing discretization and linearization errors for monotone nonlinear problems
TL;DR: In this article, the authors derived a posteriori error estimates for a class of second-order monotone quasi-linear diffusion-type problems approximated by piecewise affine, continuous finite elements.
Journal ArticleDOI
A posteriori error analysis for higher order dissipative methods for evolution problems
TL;DR: A posteriori error estimates for time discretizations by the discontinuous Galerkin method dG (q) and the corresponding implicit Runge- Kutta-Radau method IRK-R(q) of arbitrary order q≥0 are proved and conditional a posteriori errors for the time-dependent minimal surface problem are derived.
Journal ArticleDOI
A constrained integration (CINT) approach to solving partial differential equations using artificial neural networks
Keith Rudd,Silvia Ferrari +1 more
TL;DR: The CINT method combines classical Galerkin methods with a constrained backpropogation training approach to obtain an artificial neural network representation of the PDE solution that approximately satisfies the boundary conditions at every integration step.
References
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Book
A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques
TL;DR: Introduction.
Journal ArticleDOI
Mathematical Analysis and Numerical Methods for Science and Technology
TL;DR: These six volumes as mentioned in this paper compile the mathematical knowledge required by researchers in mechanics, physics, engineering, chemistry and other branches of application of mathematics for the theoretical and numerical resolution of physical models on computers.
Mathematical analysis and numerical methods for science and technology
Abstract: These six volumes - the result of a ten year collaboration between the authors, two of France's leading scientists and both distinguished international figures - compile the mathematical knowledge required by researchers in mechanics, physics, engineering, chemistry and other branches of application of mathematics for the theoretical and numerical resolution of physical models on computers. Since the publication in 1924 of the Methoden der mathematischen Physik by Courant and Hilbert, there has been no other comprehensive and up-to-date publication presenting the mathematical tools needed in applications of mathematics in directly implementable form. The advent of large computers has in the meantime revolutionised methods of computation and made this gap in the literature intolerable: the objective of the present work is to fill just this gap. Many phenomena in physical mathematics may be modeled by a system of partial differential equations in distributed systems: a model here means a set of equations, which together with given boundary data and, if the phenomenon is evolving in time, initial data, defines the system. The advent of high-speed computers has made it possible for the first time to caluclate values from models accurately and rapidly. Researchers and engineers thus have a crucial means of using numerical results to modify and adapt arguments and experiments along the way. Every fact of technical and industrial activity has been affected by these developments. Modeling by distributed systems now also supports work in many areas of physics (plasmas, new materials, astrophysics, geophysics), chemistry and mechanics and is finding increasing use in the life sciences. Volumes 5 and 6 cover problems of Transport and Evolution.
Journal ArticleDOI
Finite element interpolation of nonsmooth functions satisfying boundary conditions
L. Ridgway Scott,Shangyou Zhang +1 more
TL;DR: In this article, a modified Lagrange type interpolation operator is proposed to approximate functions in Sobolev spaces by continuous piecewise polynomials, and the combination of averaging and interpolation is shown to be a projection, and optimal error estimates are proved for the projection error.
Journal ArticleDOI
Error Estimates for Adaptive Finite Element Computations
I. Babuvška,W. C. Rheinboldt +1 more
TL;DR: The main theorem gives an error estimate in terms of localized quantities which can be computed approximately, and the estimate is optimal in the sense that, up to multiplicative constants which are independent of the mesh and solution, the upper and lower error bounds are the same.