scholarly journals Analysis of Backward Euler Primal DPG Methods

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Thomas Führer ◽  
Norbert Heuer ◽  
Michael Karkulik
Keyword(s):  
Error Bounds ◽  
Field Variable ◽  
Parabolic Problems ◽  
Optimal Error Estimates ◽  
Projection Operators ◽  
Time Step ◽  
Optimal Error ◽  
Time Stepping ◽  
Backward Euler ◽  
Elliptic Projection

Abstract We analyze backward Euler time stepping schemes for a primal DPG formulation of a class of parabolic problems. Optimal error estimates are shown in a natural norm and in the L 2 {L^{2}} norm of the field variable. For the heat equation the solution of our primal DPG formulation equals the solution of a standard Galerkin scheme and, thus, optimal error bounds are found in the literature. In the presence of advection and reaction terms, however, the latter identity is not valid anymore and the analysis of optimal error bounds requires to resort to elliptic projection operators. It is essential that these operators be projections with respect to the spatial part of the PDE, as in standard Galerkin schemes, and not with respect to the full PDE at a time step, as done previously.

A Discontinuous Finite Volume Element Method Based on Bilinear Trial Functions

2017 ◽  
Vol 14 (03) ◽  
pp. 1750025 ◽  
Author(s):  
Tie Zhang ◽  
Lixin Tang
Keyword(s):  
Finite Volume ◽  
Volume Element ◽  
Parabolic Problems ◽  
Optimal Error Estimates ◽  
Optimal Error ◽  
Finite Volume Element ◽  
The Stability ◽  
Bilinear Functions ◽  
First Time ◽  
Optimal Formula

We propose a discontinuous finite volume element (DFVE) method for second order elliptic and parabolic problems. Discontinuous bilinear functions are used as the trial functions. We give the stability analysis of this DFVE method and derive the optimal error estimates in the broken [Formula: see text]-norm. Specifically, the optimal [Formula: see text]-error is obtained for the first time for the bilinear DFVE methods solving elliptic and parabolic problems.

scholarly journals Energy-corrected FEM and explicit time-stepping for parabolic problems

2019 ◽  
Vol 53 (6) ◽  
pp. 1893-1914
Author(s):  
Piotr Swierczynski ◽  
Barbara Wohlmuth
Keyword(s):  
Finite Element ◽  
Finite Element Approximation ◽  
Regularity Of Solutions ◽  
Parabolic Problems ◽  
Element Approximation ◽  
Computational Domain ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Time Stepping

The presence of corners in the computational domain, in general, reduces the regularity of solutions of parabolic problems and diminishes the convergence properties of the finite element approximation introducing a so-called “pollution effect”. Standard remedies based on mesh refinement around the singular corner result in very restrictive stability requirements on the time-step size when explicit time integration is applied. In this article, we introduce and analyse the energy-corrected finite element method for parabolic problems, which works on quasi-uniform meshes, and, based on it, create fast explicit time discretisation. We illustrate these results with extensive numerical investigations not only confirming the theoretical results but also showing the flexibility of the method, which can be applied in the presence of multiple singular corners and a three-dimensional setting. We also propose a fast explicit time-stepping scheme based on a piecewise cubic energy-corrected discretisation in space completed with mass-lumping techniques and numerically verify its efficiency.

scholarly journals Decoupled Scheme for Time-Dependent Natural Convection Problem II: Time Semidiscreteness

2014 ◽  
Vol 2014 ◽  
pp. 1-23 ◽  
Author(s):  
Tong Zhang ◽  
ZhenZhen Tao
Keyword(s):  
Natural Convection ◽  
Error Estimates ◽  
Numerical Scheme ◽  
Computational Cost ◽  
Time Dependent ◽  
Optimal Error Estimates ◽  
Optimal Error ◽  
Backward Euler ◽  
Convection Problem ◽  
Natural Convection Problem

We study the numerical methods for time-dependent natural convection problem that models coupled fluid flow and temperature field. A coupled numerical scheme is analyzed for the considered problem based on the backward Euler scheme; stability and the corresponding optimal error estimates are presented. Furthermore, a decoupled numerical scheme is proposed by decoupling the nonlinear terms via temporal extrapolation; optimal error estimates are established. Finally, some numerical results are provided to verify the performances of the developed algorithms. Compared with the coupled numerical scheme, the decoupled algorithm not only keeps good accuracy but also saves a lot of computational cost. Both theoretical analysis and numerical experiments show the efficiency and effectiveness of the decoupled method for time-dependent natural convection problem.

scholarly journals A Time-Stepping DPG Scheme for the Heat Equation

2017 ◽  
Vol 17 (2) ◽  
pp. 237-252 ◽  
Author(s):  
Thomas Führer ◽  
Norbert Heuer ◽  
Jhuma Sen Gupta
Keyword(s):  
Heat Equation ◽  
Order Approximation ◽  
Test Functions ◽  
Optimal Test ◽  
Time Step ◽  
Time Stepping ◽  
Backward Euler ◽  
The Stability ◽  
Convergence Orders ◽  
Theoretical Results

AbstractWe introduce and analyze a discontinuous Petrov–Galerkin method with optimal test functions for the heat equation. The scheme is based on the backward Euler time stepping and uses an ultra-weak variational formulation at each time step. We prove the stability of the method for the field variables (the original unknown and its gradient weighted by the square root of the time step) and derive a Céa-type error estimate. For low-order approximation spaces this implies certain convergence orders when time steps are not too small in comparison with mesh sizes. Some numerical experiments are reported to support our theoretical results.

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