How many jumps? Variational characterization of the limit solution of a singular perturbation problem

1980 ◽  
pp. 159-180 ◽  
Author(s):  
O. Diekmann ◽  
D. Hilhorst
Keyword(s):  
Singular Perturbation ◽  
Singular Perturbation Problem ◽  
Variational Characterization ◽  
Perturbation Problem ◽  
Limit Solution

A SINGULAR PERTURBATION PROBLEM IN FRACTURED MEDIA WITH PARALLEL DIFFUSION

1998 ◽  
Vol 08 (04) ◽  
pp. 645-655 ◽  
Author(s):  
J. DAVID LOGAN ◽  
GLENN LEDDER ◽  
MICHELLE REEB HOMP
Keyword(s):  
Boundary Condition ◽  
Singular Perturbation ◽  
Porous Matrix ◽  
Fractured Media ◽  
Singular Perturbation Problem ◽  
Single Fracture ◽  
Direction Parallel ◽  
Flux Boundary Condition ◽  
Perturbation Problem ◽  
Parallel Diffusion

We study differential equations that model contaminant flow in a semi-infinite, fractured, porous medium consisting of a single fracture channel bounded by a porous matrix. Models in the literature usually do not incorporate diffusion in the porous matrix in the direction parallel to the fracture, and therefore they must omit a no-flux boundary condition at the edge, which, in some problems, may be unphysical. Herein we show that the problem usually treated in the literature is the outer problem for a correctly posed singular perturbation problem which includes diffusion in both directions as well as the no-flux boundary condition.

AC0-nonconforming quadrilateral finite element for the fourth-order elliptic singular perturbation problem

2018 ◽  
Vol 52 (5) ◽  
pp. 1981-2001 ◽  
Author(s):  
Yuan Bao ◽  
Zhaoliang Meng ◽  
Zhongxuan Luo
Keyword(s):  
Singular Perturbation ◽  
Fourth Order ◽  
Degrees Of Freedom ◽  
Perturbation Parameter ◽  
Singular Perturbation Problem ◽  
Mean Values ◽  
Perturbation Problem ◽  
Convex Quadrilateral ◽  
The Mean ◽  
Quadrilateral Finite Element

In this paper, aC0nonconforming quadrilateral element is proposed to solve the fourth-order elliptic singular perturbation problem. For each convex quadrilateralQ, the shape function space is the union ofS21(Q*) and a bubble space. The degrees of freedom are defined by the values at vertices and midpoints on the edges, and the mean values of integrals of normal derivatives over edges. The local basis functions of our element can be expressed explicitly by a new reference quadrilateral rather than by solving a linear system. It is shown that the method converges uniformly in the perturbation parameter. Lastly, numerical tests verify the convergence analysis.

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