1,720,998 research outputs found
An Efficient Linear Scheme to Approximate Parabolic Free Free Boundary Problems: Error Estimates and Implementation
No full textNochetto, R.; Verdi, C.. (1987). An Efficient Linear Scheme to Approximate Parabolic Free Free Boundary Problems: Error Estimates and Implementation. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/4538
Optimal L-infty-error estimates for nonconforming and mixed finite element methods of lowest order
No full text
Sharp maximum norm error estimates for general mixed finite element approximations to second order elliptic equations
Full text linkNonlinear evolution governed by accretive operators in Banach spaces: error control and applications
No full textNonlinear evolution equations governed by m-accretive operators in
Banach spaces are discretized via the backward or forward Euler methods with variable stepsize. Computable a posteriori error estimates are derived in terms of the discrete solution and data, and shown to converge with optimal order O(√τ ).
Applications to scalar conservation laws and degenerate parabolic equations (with or without hysteresis) in L^1, as well as to Hamilton-Jacobi equations in C^0 are given.
The error analysis relies on a comparison principle, for the novel notion of relaxed solutions, which combines and simplifies techniques of Benilan and Kruzkov. Our results provide a unified framework for existence, uniqueness and error analysis, and yield a new proof of the celebrated Crandall-Liggett error estimate
A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations
No full textWe study the backward Euler method with variable time-steps for abstract evolution equations in Hilbert spaces. Exploiting convexity of the underlying potential or the angle-bounded condition, thereby assuming no further regularity, we derive novel a posteriori estimates of the discretization error in terms of computable quantities related
to the amount of energy dissipation or monotonicity residual. These estimators solely depend on the discrete solution and data and impose no constraints between consecutive time-steps. We also prove that they converge to zero with an optimal rate with respect to the regularity of the solution. We apply the abstract results to a number of concrete strongly nonlinear problems of parabolic type with degenerate or singular character
Adaptive Fourier-Galerkin methods
No full textWe study the performance of adaptive Fourier-Galerkin methods in a periodic box in R^d with dimension d>1. These methods offer unlimited approximation power only restricted by solution and data regularity. They are of intrinsic interest but are also a first step towards understanding adaptivity for the hp-FEM. We examine two nonlinear approximation classes, one classical corresponding to algebraic decay of Fourier coefficients and another associated with exponential decay typical of spectral approximation. We investigate the natural sparsity class for the operator range and find that the exponential class is not preserved, thus in contrast with the algebraic class. This entails a striking different behavior of the feasible residuals that lead to practical algorithms, influencing the overall optimality. The sparsity degradation for the exponential class is partially compensated with coarsening. We present several feasible adaptive Fourier algorithms, prove their contraction properties, and examine the cardinality of the activated sets. The Galerkin approximations at the end of each iteration are quasi-optimal for both classes, but inner loops or intermediate approximations are sub-optimal for the exponential class
A safeguarded dual weighted residual method
No full textThe dual weighted residual (DWR) method yields reliable a posteriori error bounds for linear output functionals provided that the error incurred by the numerical approximation of the dual solution is negligible. In that case, its performance is generally superior than that of global ‘energy norm’ error estimators which are ‘unconditionally’ reliable. We present a simple numerical example for which neglecting the approximation error leads to severe underestimation of the functional error, thus showing that the DWR method may be unreliable. We propose a remedy that preserves the original performance, namely a DWR method safeguarded by additional asymptotically higher order a posteriori terms. In particular, the enhanced estimator is unconditionally reliable and asymptotically coincides with the original DWR method. These properties are illustrated via the aforementioned example
Error control of nonlinear evolution equations
No full textWe derive novel a posteriori error estimates for backward Euler approximations of evo- lution inequalities in Hilbert spaces. The underlying nonlinear (multivalued) monotone operator is subdifferential, or more generally angle-bounded. The estimates depend solely on the discrete so- lution data, impose no constraints between consecutive time-steps, exhibit explicit stability factors, and are optimal with respect to both order and regularity
- …
