Applications of approximate gradient schemes for nonlinear parabolic equations

• Published in: Applications of mathematics (Prague) Vol. 60; no. 2; pp. 135 - 156
• Main Authors: Eymard, Robert, Handlovičová, Angela, Herbin, Raphaèle, Mikula, Karol, Stašová, Olga
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-04-2015; Springer Nature B.V; Akademie věd České republiky, Matematický ústav
• Subjects: Analysis; Applications of Mathematics; Approximation; Classical and Continuum Physics; Convergence; Diffusion; Eigenvalues; Eigenvectors; Filtering; Hypotheses; Mathematical analysis; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical models; Mathematics; Mathematics and Statistics; Nonlinear equations; Nonlinearity; Numerical Analysis; Optimization; Studies; Theorems; Theoretical; regularized Perona-Malik equation; 65M08; 65M12; gradient schemes
• ISSN: 0862-7940; 1572-9109
• DOI: 10.1007/s10492-015-0088-4

We develop gradient schemes for the approximation of the Perona-Malik equations and nonlinear tensor-diffusion equations. We prove the convergence of these methods to the weak solutions of the corresponding nonlinear PDEs. A particular gradient scheme on rectangular meshes is then studied numerically with respect to experimental order of convergence which shows its second order accuracy. We present also numerical experiments related to image filtering by time-delayed Perona-Malik and tensor diffusion equations.