Linear scheme for finite element solution of nonlinear parabolic-elliptic problems with nonhomogeneous Dirichlet boundary condition

Dana Říhová-Škabrahová

Applications of Mathematics (2001)

  • Volume: 46, Issue: 2, page 103-144
  • ISSN: 0862-7940

Abstract

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The computation of nonlinear quasistationary two-dimensional magnetic fields leads to a nonlinear second order parabolic-elliptic initial-boundary value problem. Such a problem with a nonhomogeneous Dirichlet boundary condition on a part Γ 1 of the boundary is studied in this paper. The problem is discretized in space by the finite element method with linear functions on triangular elements and in time by the implicit-explicit method (the left-hand side by the implicit Euler method and the right-hand side by the explicit Euler method). The scheme we get is linear. The strong convergence of the method is proved under the assumptions that the boundary ∂ Ω is piecewise of class C 3 and the initial condition belongs to L 2 only. Strong monotonicity and Lipschitz continuity of the form a ( v , w ) is not an assumption, but a property of this form following from its physical background.

How to cite

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Říhová-Škabrahová, Dana. "Linear scheme for finite element solution of nonlinear parabolic-elliptic problems with nonhomogeneous Dirichlet boundary condition." Applications of Mathematics 46.2 (2001): 103-144. <http://eudml.org/doc/33079>.

@article{Říhová2001,
abstract = {The computation of nonlinear quasistationary two-dimensional magnetic fields leads to a nonlinear second order parabolic-elliptic initial-boundary value problem. Such a problem with a nonhomogeneous Dirichlet boundary condition on a part $\Gamma \!_1$ of the boundary is studied in this paper. The problem is discretized in space by the finite element method with linear functions on triangular elements and in time by the implicit-explicit method (the left-hand side by the implicit Euler method and the right-hand side by the explicit Euler method). The scheme we get is linear. The strong convergence of the method is proved under the assumptions that the boundary $\partial \Omega $ is piecewise of class $C^3$ and the initial condition belongs to $L_2$ only. Strong monotonicity and Lipschitz continuity of the form $a(v,w)$ is not an assumption, but a property of this form following from its physical background.},
author = {Říhová-Škabrahová, Dana},
journal = {Applications of Mathematics},
keywords = {finite element method; parabolic-elliptic problems; two-dimensional electromagnetic field; parabolic-elliptic problems; finite element method; two-dimensional electromagnetic field; Dirichlet problem; quasistationary magnetic field; implicit-explicit method; convergence},
language = {eng},
number = {2},
pages = {103-144},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Linear scheme for finite element solution of nonlinear parabolic-elliptic problems with nonhomogeneous Dirichlet boundary condition},
url = {http://eudml.org/doc/33079},
volume = {46},
year = {2001},
}

TY - JOUR
AU - Říhová-Škabrahová, Dana
TI - Linear scheme for finite element solution of nonlinear parabolic-elliptic problems with nonhomogeneous Dirichlet boundary condition
JO - Applications of Mathematics
PY - 2001
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 46
IS - 2
SP - 103
EP - 144
AB - The computation of nonlinear quasistationary two-dimensional magnetic fields leads to a nonlinear second order parabolic-elliptic initial-boundary value problem. Such a problem with a nonhomogeneous Dirichlet boundary condition on a part $\Gamma \!_1$ of the boundary is studied in this paper. The problem is discretized in space by the finite element method with linear functions on triangular elements and in time by the implicit-explicit method (the left-hand side by the implicit Euler method and the right-hand side by the explicit Euler method). The scheme we get is linear. The strong convergence of the method is proved under the assumptions that the boundary $\partial \Omega $ is piecewise of class $C^3$ and the initial condition belongs to $L_2$ only. Strong monotonicity and Lipschitz continuity of the form $a(v,w)$ is not an assumption, but a property of this form following from its physical background.
LA - eng
KW - finite element method; parabolic-elliptic problems; two-dimensional electromagnetic field; parabolic-elliptic problems; finite element method; two-dimensional electromagnetic field; Dirichlet problem; quasistationary magnetic field; implicit-explicit method; convergence
UR - http://eudml.org/doc/33079
ER -

References

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