scholarly journals Combining the DPG Method with Finite Elements

2018 ◽  
Vol 18 (4) ◽  
pp. 639-652
Author(s):  
Thomas Führer ◽  
Norbert Heuer ◽  
Michael Karkulik ◽  
Rodolfo Rodríguez
Keyword(s):  
Variational Formulation ◽  
Model Problem ◽  
Standard Test ◽  
Well Posedness ◽  
Bounded Domains ◽  
Test Functions ◽  
Test Space ◽  
Optimal Test ◽  
Weak Part ◽  
Convergence Orders

AbstractWe propose and analyze a discretization scheme that combines the discontinuous Petrov–Galerkin and finite element methods. The underlying model problem is of general diffusion-advection-reaction type on bounded domains, with decomposition into two sub-domains. We propose a heterogeneous variational formulation that is of the ultra-weak (Petrov–Galerkin) form with broken test space in one part, and of Bubnov–Galerkin form in the other. A standard discretization with conforming approximation spaces and appropriate test spaces (optimal test functions for the ultra-weak part and standard test functions for the Bubnov–Galerkin part) gives rise to a coupled DPG-FEM scheme. We prove its well-posedness and quasi-optimal convergence. Numerical results confirm expected convergence orders.

Alternative Enriched Test Spaces in the DPG Method for Singular Perturbation Problems

2019 ◽  
Vol 19 (3) ◽  
pp. 603-630 ◽  
Author(s):  
Jacob Salazar ◽  
Jaime Mora ◽  
Leszek Demkowicz
Keyword(s):  
Singular Perturbation ◽  
Perturbation Parameter ◽  
Standard Test ◽  
Test Functions ◽  
Optimal Test ◽  
Test Norms ◽  
Local Test ◽  
Linear Problems ◽  
Perturbation Problems ◽  
Convection Dominated

AbstractWe propose and investigate the application of alternative enriched test spaces in the discontinuous Petrov–Galerkin (DPG) finite element framework for singular perturbation linear problems, with an emphasis on 2D convection-dominated diffusion. Providing robust {L^{2}} error estimates for the field variables is considered a convenient feature for this class of problems, since this norm would not account for the large gradients present in boundary layers. With this requirement in mind, Demkowicz and others have previously formulated special test norms, which through DPG deliver the desired {L^{2}} convergence. However, robustness has only been verified through numerical experiments for tailored test norms which are problem-specific, whereas the quasi-optimal test norm (not problem specific) has failed such tests due to the difficulty to resolve the optimal test functions sought in the DPG technology. To address this issue (i.e. improve optimal test functions resolution for the quasi-optimal test norm), we propose to discretize the local test spaces with functions that depend on the perturbation parameter ϵ. Explicitly, we work with B-spline spaces defined on an ϵ-dependent Shishkin submesh. Two examples are run using adaptive h-refinement to compare the performance of proposed test spaces with that of standard test spaces. We also include a modified norm and a continuation strategy aiming to improve time performance and briefly experiment with these ideas.

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.

scholarly journals The BV solution of the parabolic equation with degeneracy on the boundary

2016 ◽  
Vol 14 (1) ◽  
pp. 272-282
Author(s):  
Huashui Zhan ◽  
Shuping Chen
Keyword(s):  
Boundary Condition ◽  
Parabolic Equation ◽  
Well Posedness ◽  
Test Functions ◽  
The Stability

AbstractConsider a parabolic equation which is degenerate on the boundary. By the degeneracy, to assure the well-posedness of the solutions, only a partial boundary condition is generally necessary. When 1 ≤ α < p – 1, the existence of the local BV solution is proved. By choosing some kinds of test functions, the stability of the solutions based on a partial boundary condition is established.

scholarly journals The State of Art on Super Structural Testing Machine in China

2017 ◽  
Vol 11 (1) ◽  
pp. 1013-1023
Author(s):  
Yubin Ma ◽  
Jinbao Ji ◽  
Weiming Yan ◽  
Jing Han
Keyword(s):  
Testing Machine ◽  
Structural Testing ◽  
Test Functions ◽  
Test Space ◽  
Test Pieces ◽  
University Of Technology ◽  
Loading System ◽  
Further Development ◽  
Test Loading ◽  
Structure Test

As important test equipment to carry out full-scale structure testing and components failure mechanisms research, super structural testing machines have developed rapidly all over the world in recent years. The authors of this paper investigated, compared and analyzed the construction and the development trends of super structural testing machines in China. The test functions, the test space, the maximum tonnage and other technical indexes of super structural testing machines, which have been built in China, were described. The functions and the typical tests of a multi-functional structure test loading system (40,000 kN) installed in the Beijing University of Technology were also discussed. The vibrations caused by the failure of brittle test pieces on this super structural testing machine were analyzed, and some suggestions to reduce the vibrations were given. It would be valuable for further development and the improvement of other super structural testing machines.

Well Posedness

2012 ◽  
Author(s):  
G. F. Roach ◽  
I. G. Stratis ◽  
A. N. Yannacopoulos
Keyword(s):  
Variational Formulation ◽  
Eigenvalue Problems ◽  
Dimensional Space ◽  
Well Posedness ◽  
Finite Dimensional Space ◽  
Chiral Media ◽  
Exterior Problems ◽  
Finite Dimensional ◽  
Time Harmonic ◽  
Interior Domain

This chapter presents rigorous mathematical results concerning the solvability and well posedness of time-harmonic problems for complex electromagnetic media, with a special emphasis on chiral media. It also presents some results concerning eigenvalue problems in cavities filled with complex electromagnetic materials. The chapter also studies the behaviour of the interior domain problem for a chiral medium in the limit of low chirality. Next, it presents some comments related to the well posedness and solvability of exterior problems. Finally, using an appropriate finite-dimensional space and the variational formulation of the discretised version of the original boundary value problem, this chapter obtains numerical methods for the solution of the Maxwell equations for chiral media.

scholarly journals Hyperbolic Maxwell variational inequalities of the second kind

2020 ◽  
Vol 26 ◽  
pp. 34 ◽  
Author(s):  
Irwin Yousept
Keyword(s):  
Variational Inequalities ◽  
Existence Result ◽  
Semigroup Theory ◽  
Regularity Property ◽  
Well Posedness ◽  
Test Functions ◽  
Local Boundedness ◽  
Subdifferential Calculus ◽  
Faraday’S Law ◽  
Yosida Regularization

We analyze a class of hyperbolic Maxwell variational inequalities of the second kind. By means of a local boundedness assumption on the subdifferential of the underlying nonlinearity, we prove a well-posedness result, where the main tools for the proof are the semigroup theory for Maxwell’s equations, the Yosida regularization and the subdifferential calculus. The second part of the paper focuses on a more general case omitting the local boundedness assumption. In this case, taking into account more regular initial data and test functions, we are able to prove a weaker existence result through the use of the minimal section operator associated with the Nemytskii operator of the governing subdifferential. Eventually, we transfer the developed well-posedness results to the case involving Faraday’s law, which in particular allows us to improve the regularity property of the electric field in the weak existence result.

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