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Construction of convergent adaptive weighted essentially non-oscillatory schemes for Hamilton-Jacobi equations on triangular meshes

Kwangil Kim, Unhyok Hong, Kwanhung Ri, Juhyon Yu (2021)

Applications of Mathematics

We propose a method of constructing convergent high order schemes for Hamilton-Jacobi equations on triangular meshes, which is based on combining a high order scheme with a first order monotone scheme. According to this methodology, we construct adaptive schemes of weighted essentially non-oscillatory type on triangular meshes for nonconvex Hamilton-Jacobi equations in which the first order monotone approximations are occasionally applied near singular points of the solution (discontinuities of...

Discontinuous Galerkin method for nonlinear convection-diffusion problems with mixed Dirichlet-Neumann boundary conditions

Oto Havle, Vít Dolejší, Miloslav Feistauer (2010)

Applications of Mathematics

The paper is devoted to the analysis of the discontinuous Galerkin finite element method (DGFEM) applied to the space semidiscretization of a nonlinear nonstationary convection-diffusion problem with mixed Dirichlet-Neumann boundary conditions. General nonconforming meshes are used and the NIPG, IIPG and SIPG versions of the discretization of diffusion terms are considered. The main attention is paid to the impact of the Neumann boundary condition prescribed on a part of the boundary on the truncation...

Error Control and Andaptivity for a Phase Relaxation Model

Zhiming Chen, Ricardo H. Nochetto, Alfred Schmidt (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The phase relaxation model is a diffuse interface model with small parameter ε which consists of a parabolic PDE for temperature θ and an ODE with double obstacles for phase variable χ. To decouple the system a semi-explicit Euler method with variable step-size τ is used for time discretization, which requires the stability constraint τ ≤ ε. Conforming piecewise linear finite elements over highly graded simplicial meshes with parameter h are further employed for space discretization. A posteriori...

Error estimates for barycentric finite volumes combined with nonconforming finite elements applied to nonlinear convection-diffusion problems

Vít Dolejší, Miloslav Feistauer, Jiří Felcman, Alice Kliková (2002)

Applications of Mathematics

The subject of the paper is the derivation of error estimates for the combined finite volume-finite element method used for the numerical solution of nonstationary nonlinear convection-diffusion problems. Here we analyze the combination of barycentric finite volumes associated with sides of triangulation with the piecewise linear nonconforming Crouzeix-Raviart finite elements. Under some assumptions on the regularity of the exact solution, the L 2 ( L 2 ) and L 2 ( H 1 ) error estimates are established. At the end...

Facilitating the Adoption of Unstructured High-Order Methods Amongst a Wider Community of Fluid Dynamicists

P. E. Vincent, A. Jameson (2011)

Mathematical Modelling of Natural Phenomena

Theoretical studies and numerical experiments suggest that unstructured high-order methods can provide solutions to otherwise intractable fluid flow problems within complex geometries. However, it remains the case that existing high-order schemes are generally less robust and more complex to implement than their low-order counterparts. These issues, in conjunction with difficulties generating high-order meshes, have limited the adoption of high-order...

Fully adaptive multiresolution schemes for strongly degenerate parabolic equations in one space dimension

Raimund Bürger, Ricardo Ruiz, Kai Schneider, Mauricio Sepúlveda (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

We present a fully adaptive multiresolution scheme for spatially one-dimensional quasilinear strongly degenerate parabolic equations with zero-flux and periodic boundary conditions. The numerical scheme is based on a finite volume discretization using the Engquist-Osher numerical flux and explicit time stepping. An adaptive multiresolution scheme based on cell averages is then used to speed up the CPU time and the memory requirements of the underlying finite volume scheme, whose first-order...

Galerkin time-stepping methods for nonlinear parabolic equations

Georgios Akrivis, Charalambos Makridakis (2004)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We consider discontinuous as well as continuous Galerkin methods for the time discretization of a class of nonlinear parabolic equations. We show existence and local uniqueness and derive optimal order optimal regularity a priori error estimates. We establish the results in an abstract Hilbert space setting and apply them to a quasilinear parabolic equation.

Galerkin time-stepping methods for nonlinear parabolic equations

Georgios Akrivis, Charalambos Makridakis (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider discontinuous as well as continuous Galerkin methods for the time discretization of a class of nonlinear parabolic equations. We show existence and local uniqueness and derive optimal order optimal regularity a priori error estimates. We establish the results in an abstract Hilbert space setting and apply them to a quasilinear parabolic equation.

Gaussian curvature based tangential redistribution of points on evolving surfaces

Medľa, Matej, Mikula, Karol (2017)

Proceedings of Equadiff 14

There exist two main methods for computing a surface evolution, level-set method and Lagrangian method. Redistribution of points is a crucial element in a Lagrangian approach. In this paper we present a point redistribution that compress quads in the areas with a high Gaussian curvature. Numerical method is presented for a mean curvature flow of a surface approximated by quads.

Genuinely multi-dimensional non-dissipative finite-volume schemes for transport

Bruno Després, Frédéric Lagoutière (2007)

International Journal of Applied Mathematics and Computer Science

We develop a new multidimensional finite-volume algorithm for transport equations. This algorithm is both stable and non-dissipative. It is based on a reconstruction of the discrete solution inside each cell at every time step. The proposed reconstruction, which is genuinely multidimensional, allows recovering sharp profiles in both the direction of the transport velocity and the transverse direction. It constitutes an extension of the one-dimensional reconstructions analyzed in (Lagoutière, 2005;...

Grid adjustment based on a posteriori error estimators

Karel Segeth (1993)

Applications of Mathematics

The adjustment of one-dimensional space grid for a parabolic partial differential equation solved by the finite element method of lines is considered in the paper. In particular, the approach based on a posteriori error indicators and error estimators is studied. A statement on the rate of convergence of the approximation of error by estimator to the error in the case of a system of parabolic equations is presented.

Maximum-norm resolvent estimates for elliptic finite element operators on nonquasiuniform triangulations

Nikolai Yu. Bakaev, Michel Crouzeix, Vidar Thomée (2006)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In recent years several papers have been devoted to stability and smoothing properties in maximum-norm of finite element discretizations of parabolic problems. Using the theory of analytic semigroups it has been possible to rephrase such properties as bounds for the resolvent of the associated discrete elliptic operator. In all these cases the triangulations of the spatial domain has been assumed to be quasiuniform. In the present paper we show a resolvent estimate, in one and two space dimensions,...

Maximum-norm resolvent estimates for elliptic finite element operators on nonquasiuniform triangulations

Nikolai Yu. Bakaev, Michel Crouzeix, Vidar Thomée (2007)

ESAIM: Mathematical Modelling and Numerical Analysis


In recent years several papers have been devoted to stability and smoothing properties in maximum-norm of finite element discretizations of parabolic problems. Using the theory of analytic semigroups it has been possible to rephrase such properties as bounds for the resolvent of the associated discrete elliptic operator. In all these cases the triangulations of the spatial domain has been assumed to be quasiuniform. In the present paper we show a resolvent estimate, in one and two space dimensions, under...

Modelling of natural convection flows with large temperature differences : a benchmark problem for low Mach number solvers. Part 1. Reference solutions

Patrick Le Quéré, Catherine Weisman, Henri Paillère, Jan Vierendeels, Erik Dick, Roland Becker, Malte Braack, James Locke (2005)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

There are very few reference solutions in the literature on non-Boussinesq natural convection flows. We propose here a test case problem which extends the well-known De Vahl Davis differentially heated square cavity problem to the case of large temperature differences for which the Boussinesq approximation is no longer valid. The paper is split in two parts: in this first part, we propose as yet unpublished reference solutions for cases characterized by a non-dimensional temperature difference of...

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