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The Discontinuity Problem [article]

Vasco Brattka
2020 arXiv   pre-print
discontinuity of the problem.  ...  We introduce the discontinuity problem, and we show that it is reducible exactly to the effectively discontinuous problems, defined in a suitable way.  ...  Likewise, we would like to thank Arno Pauly for a discussion on Wadge games for problems that has helped to improve the corresponding results.  ... 
arXiv:2012.02143v1 fatcat:f4k4vg6j3fb73fvhyx546wp2uq

ON THE DISCONTINUOUS RIEMANN-HILBERT PROBLEM

Henryk Łubowicz, Bohdan Wieprzkowicz
1970 Demonstratio Mathematica  
FORMULATION OP THE PROBLEM By the Riemann-Hilbert non-linear discontinuous problem we understand the following problem.  ...  The functions w% u" , p.*, </>* are solutions of the system (34). So the function w* is a solution of the discontinuous generalized Riemann-Hilbert problem in the multiconnected domain.  ...  On the discontinuous Riemann-Hilbert problem  ... 
doi:10.1515/dema-1970-0304 fatcat:ydfljfcyjvaexozdlfblx54u6m

The Compact Discontinuous Galerkin (CDG) Method for Elliptic Problems [article]

Jaume Peraire, Per-Olof Persson
2008 arXiv   pre-print
We present a compact discontinuous Galerkin (CDG) method for an elliptic model problem.  ...  The problem is first cast as a system of first order equations by introducing the gradient of the primal unknown, or flux, as an additional variable.  ...  Discontinuous Galerkin formulation 2.1 Problem definition The proposed method will be described for the model Poisson problem −∇ · (κ∇u) = f in Ω, u = g D on ∂Ω D , κ ∂u ∂n = g N on ∂Ω N , (1) where Ω  ... 
arXiv:math/0702353v3 fatcat:lyljxf2h6rfkximltzem5rilre

Adaptively enriched coarse space for the discontinuous Galerkin multiscale problems [article]

Erik Eikeland, Leszek Marcinkowski, Talal Rahman
2018 arXiv   pre-print
In this paper, we propose a two-level overlapping additive Schwarz domain decomposition preconditioner for the symmetric interior penalty discontinuous Galerkin method for the second order elliptic boundary  ...  value problem with highly heterogeneous coefficients.  ...  The objective of this paper is to further extend this idea to the discontinuous Galerkin case, by proposing a new and effective coarse problem for the discontinuous Galerkin method, which is based on adaptively  ... 
arXiv:1706.02325v3 fatcat:hzt56qlf3vfplay3quwxpota6i

Discontinuous Galerkin approximation of the fully-coupled thermo-poroelastic problem [article]

Paola F. Antonietti and Stefano Bonetti and Michele Botti
2022 arXiv   pre-print
We present and analyze a discontinuous Galerkin method for the numerical modelling of the non-linear fully-coupled thermo-poroelastic problem.  ...  For the spatial discretization, we design a high-order discontinuous Galerkin method on polygonal and polyhedral grids based on a novel four-field formulation of the problem.  ...  The spatial discretization of the aforementioned problem is set into the framework of the discontinuous Galerkin (DG) finite element methods.  ... 
arXiv:2205.04262v3 fatcat:bas4hms7uvgy5pcq3rmlotuuwe

Discontinuous Boundary Conditions and the Dirichlet Problem

Norbert Wiener
1923 Transactions of the American Mathematical Society  
discontinuous boundary conditions of a very general sort.  ...  Evans,t but it appears to the author that though the theory of Evans gives more detailed information concerning the character of the solutions of the Dirichlet problem in the neighborhood of the boundary  ...  discontinuous boundary conditions of a very general sort.  ... 
doi:10.2307/1989291 fatcat:xfgfkjbl2nhkjjurzjvedt5a3u

The Calibration Method for Free Discontinuity Problems [article]

G. Dal Maso
2000 arXiv   pre-print
The calibration method is used to identify some minimizers of the Mumford-Shah functional. The method is then extended to more general free discontinuity problems.  ...  The main feature of these problems is that the shape and location of the discontinuity set S u are not prescribed.  ...  Introduction In [5] De Giorgi introduced the name free discontinuity problems to denote a wide class of minimum problems for functionals of the form F (u) := Ω\Su f (x, u(x), ∇u(x))dx + Su ψ(x, u + (  ... 
arXiv:math/0006016v1 fatcat:s63to4a6wfbtragw2ljc5fxbgi

Discontinuous Galerkin methods for the biharmonic problem

E. H. Georgoulis, P. Houston
2008 IMA Journal of Numerical Analysis  
Access from the University of Nottingham repository: Abstract This work is concerned with the design and analysis of hp-version discontinuous Galerkin (DG) finite element methods for boundary-value problems  ...  Anal. 39, 5 (2001/02), 1749-1779) developed for the Poisson problem, to the design of DG methods via an appropriate choice of numerical flux functions for fourth order problems; as an example we retrieve  ...  In Section 3 we present a unified approach for the construction of discontinuous Galerkin methods for the biharmonic problem via a suitable choice of numerical flux functions.  ... 
doi:10.1093/imanum/drn015 fatcat:ggntq5soanauphqiu3mgmqu2qq

Discontinuous boundary conditions and the Dirichlet problem

Norbert Wiener
1923 Transactions of the American Mathematical Society  
discontinuous boundary conditions of a very general sort.  ...  Evans,t but it appears to the author that though the theory of Evans gives more detailed information concerning the character of the solutions of the Dirichlet problem in the neighborhood of the boundary  ...  discontinuous boundary conditions of a very general sort.  ... 
doi:10.1090/s0002-9947-1923-1501246-8 fatcat:j6vwdqaup5fu3fc2mj242cj6vy

The Calibration Method for Free Discontinuity Problems [chapter]

Gianni Dal Maso
2001 European Congress of Mathematics  
The calibration method is used to identify some minimizers of the Mumford-Shah functional. The method is then extended to more general free discontinuity problems.  ...  The main feature of these problems is that the shape and location of the discontinuity set S u are not prescribed.  ...  Introduction In [5] De Giorgi introduced the name free discontinuity problems to denote a wide class of minimum problems for functionals of the form F (u) := Ω\Su f(x, u(x), ∇u(x))dx + Su ψ(x, u + (x  ... 
doi:10.1007/978-3-0348-8268-2_18 fatcat:yewe745ijzfvtiryra4ic3dkku

Well-posedness of the linearized problem for contact MHD discontinuities [article]

Alessandro Morando, Yuri Trakhinin, Paola Trebeschi
2014 arXiv   pre-print
We study the free boundary problem for contact discontinuities in ideal compressible magnetohydrodynamics (MHD).  ...  Sobolev spaces of the linearized problem for 2D planar MHD flows.  ...  Reduced nonlinear problem in a fixed domain The function ϕ(t, x ′ ) determining the contact discontinuity surface Γ is one of the unknowns of the free boundary problem (1)/(4), (11) , (13) .  ... 
arXiv:1311.6373v2 fatcat:34upe4pywvddtno7ol44mdizaa

Analysis of a new stabilized discontinuous Galerkin method for the reaction-diffusion problem with discontinuous coefficient [article]

Zhihao Ge, Jiwei Cao
2014 arXiv   pre-print
In this paper, a new stabilized discontinuous Galerkin method within a new function space setting is introduced, which involves an extra stabilization term on the normal fluxes across the element interfaces  ...  A priori error estimates are derived, which are verified by a 2D experiment on a reaction-diffusion type model problem.  ...  The weak formulation of the problem (2.3)-(2.4) The discontinuous Galerkin method is a class of finite element methods using completely discontinuous piecewise polynomial space for the numerical solution  ... 
arXiv:1411.6333v1 fatcat:4ywuk7pjlbadpjdmr67ccggeta

Discontinuous Galerkin Methods for the Biharmonic Problem on Polygonal and Polyhedral Meshes [article]

Zhaonan Dong
2018 arXiv   pre-print
We introduce an hp-version symmetric interior penalty discontinuous Galerkin finite element method (DGFEM) for the numerical approximation of the biharmonic equation on general computational meshes consisting  ...  The key feature of the proposed method is that it employs elemental polynomial bases of total degree P_p, defined in the physical coordinate system, without requiring the mapping from a given reference  ...  Dong acknowledges funding by The Leverhulme Trust (grant no. RPG-2015-306).  ... 
arXiv:1807.07817v2 fatcat:ug2uje6m3jbnfagomcugqqdjry

The Unfitted Discontinuous Galerkin Method for Solving the EEG Forward Problem [article]

Andreas Nüßing, Carsten H. Wolters, Heinrich Brinck, Christian Engwer
2016 arXiv   pre-print
Objective: The purpose of this study is to introduce and evaluate the unfitted discontinuous Galerkin finite element method (UDG-FEM) for solving the electroencephalography (EEG) forward problem.  ...  Significance: This study shows the first application of the UDG-FEM approach to the EEG forward problem.  ...  In this paper, we present the unfitted discontinuous Galerkin finite element method (UDG-FEM) to solve the EEG forward problem.  ... 
arXiv:1601.07810v2 fatcat:eldhzy2vjjcwvhcv4fdn45qxli

Analysis of the discontinuous Galerkin method for elliptic problems on surfaces

A. Dedner, P. Madhavan, B. Stinner
2013 IMA Journal of Numerical Analysis  
We extend the discontinuous Galerkin (DG) framework to a linear second-order elliptic problem on a compact smooth connected and oriented surface.  ...  This is then verified numerically for a number of test problems.  ...  Acknowledgements This research has been supported by the British Engineering and Physical Sciences Research Council (EPSRC), Grant EP/H023364/1.  ... 
doi:10.1093/imanum/drs033 fatcat:zb425htyqrdkdacqpm57zj2cuy
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