Abstract Computable a‐posteriori error estimates for finite element solutions are derived in an asymptotic form for h → 0 where h measures the size of the elements. The approach has similarity to the residual method but differs from it in the use of norms of negative Sobolev spaces corresponding to the given bilinear (energy) form. For clarity the presentation is restricted to one‐dimensional model problems. More specifically, the source, eigenvalue, and parabolic problems are considered involving a linear, self‐adjoint operator of the second order. Generalizations to more general one‐dimensional problems are straightforward, and the results also extend to higher space dimensions; but this involves some additional considerations. The estimates can be used for a practical a‐posteriori assessment of the accuracy of a computed finite element solution, and they provide a basis for the design of adaptive finite element solvers.
Based on the theory of differential equations on manifolds, existence and uniqueness results are proved for a class of mixed systems of differential and algebraic equations as they occur in various applications. Both the autonomous and nonautonomous case are considered. Moreover, a class of algebraically incomplete systems is introduced for which existence and uniqueness results only hold on certain lower-dimensional manifolds. This class includes systems for which the application of ODE-solvers is known to lead to difficulties. Finally, some solution approach based on continuation techniques is outlined.
article Free AccessA locally parameterized continuation process Authors: Werner C. Rheinboldt Institute for Computational Mathematics and Applications, Department of Mathematics and Statistics, University of Pittsburg, Pittsburg, PA Institute for Computational Mathematics and Applications, Department of Mathematics and Statistics, University of Pittsburg, Pittsburg, PAView Profile , John V. Burkardt Institute for Computational Mathematics and Applications, Department of Mathematics and Statistics, University of Pittsburg, Pittsburg, PA Institute for Computational Mathematics and Applications, Department of Mathematics and Statistics, University of Pittsburg, Pittsburg, PAView Profile Authors Info & Claims ACM Transactions on Mathematical SoftwareVolume 9Issue 2pp 215–235https://doi.org/10.1145/357456.357460Published:01 June 1983Publication History 195citation923DownloadsMetricsTotal Citations195Total Downloads923Last 12 Months29Last 6 weeks1 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF
