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The Discontinuity Problem
[article]
2020
arXiv
pre-print
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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
1970
Demonstratio Mathematica
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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]
2008
arXiv
pre-print
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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]
2018
arXiv
pre-print
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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]
2022
arXiv
pre-print
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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
1923
Transactions of the American Mathematical Society
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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]
2000
arXiv
pre-print
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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
2008
IMA Journal of Numerical Analysis
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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
1923
Transactions of the American Mathematical Society
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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]
2001
European Congress of Mathematics
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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]
2014
arXiv
pre-print
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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]
2014
arXiv
pre-print
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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]
2018
arXiv
pre-print
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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]
2016
arXiv
pre-print
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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
2013
IMA Journal of Numerical Analysis
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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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