Space-Time Adaptive Methods for the Mixed Formulation of a Linear Parabolic Problem

2017 ◽  
Vol 74 (3) ◽  
pp. 1725-1756 ◽  
Author(s):  
Dongho Kim ◽  
Eun-Jae Park ◽  
Boyoon Seo
Keyword(s):  
Parabolic Problem ◽  
Adaptive Methods ◽  
Space Time ◽  
Mixed Formulation ◽  
Linear Parabolic Problem

Fujita exponents for a space–time weighted parabolic problem in bounded domains

2021 ◽  
Vol 62 ◽  
pp. 103358
Author(s):  
Xizheng Sun ◽  
Bingchen Liu ◽  
Fengjie Li
Keyword(s):  
Parabolic Problem ◽  
Space Time ◽  
Bounded Domains

scholarly journals Smoothing properties in multistep backward difference method and time derivative approximation for linear parabolic equations

2005 ◽  
Vol 2005 (4) ◽  
pp. 523-536
Author(s):  
Yubin Yan
Keyword(s):  
Difference Method ◽  
Parabolic Equations ◽  
Parabolic Problem ◽  
Approximation Scheme ◽  
Time Derivative ◽  
Nonsmooth Data ◽  
Smoothing Property ◽  
Linear Parabolic Equations ◽  
Linear Parabolic Problem ◽  
Data Error

A smoothing property in multistep backward difference method for a linear parabolic problem in Hilbert space has been proved, where the operator is selfadjoint, positive definite with compact inverse. By using the solutions computed by a multistep backward difference method for the parabolic problem, we introduce an approximation scheme for time derivative. The nonsmooth data error estimate for the approximation of time derivative has been obtained.

Local Space-Time Adaptive Discontinuous Galerkin Finite Element Methods for Time-Dependent Waves

2003 ◽  
Author(s):  
Lonny L. Thompson ◽  
Dantong He
Keyword(s):  
Discontinuous Galerkin ◽  
Acoustic Radiation ◽  
Adaptive Strategy ◽  
Adaptive Methods ◽  
Space Time ◽  
Time Dependent ◽  
Time Interval ◽  
Time Step ◽  
Continuous Space ◽  
Local Space

Local space-time adaptive methods are developed including high-order accurate nonreflecting boundary conditions (NRBC) for time-dependent waves. The time-discontinuous Galerkin (TDG) variational method is used to divide the time-interval into space-time slabs, the solution advanced from one slab to the next. Within each slab, a continuous space-time mesh is used which enables local sub-time steps. By maintaining orthogonality of the space-time mesh and pre-integrating analytically through the time-slab, we obtain an efficient yet robust local space-time adaptive method. Any standard spatial element may be used together with standard spatial mesh generation and visualization methods. Recovery based error estimates are used in both space and time dimensions to determine the number and size of local space-time elements within a global time step such that both the spatial and temporal estimated error is equally distributed throughout the space-time approximation. The result is an efficient and reliable adaptive strategy which distributes local space-time elements where needed to accurately track time-dependent waves over large distances and time. Numerical examples of time-dependent acoustic radiation are given which demonstrate the accuracy, reliability and efficiency gained from this new technology.

Sign in / Sign up

Export Citation Format

Share Document