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We give details of the theory of primal domain decomposition (DD) methods for a 2-dimensional second order elliptic equation with homogeneous Dirichlet boundary conditions and jumping coefficients. The problem is discretized by the finite element method. The computational domain is decomposed into triangular subdomains that align with the coefficients jumps. We prove that the condition number of the vertex-based DD preconditioner is , independently of the coefficient jumps, where and denote...
This paper deals with the solution of problems involving partial differential
equations in . For three dimensional case, methods are useful if they
require neither domain boundary regularity nor regularity for the exact solution of
the problem. A new domain decomposition method is therefore presented which
uses low degree finite elements. The numerical approximation of the
solution is easy, and optimal error bounds are obtained according to suitable
norms.
The aim of this work is to compare a new uncoupled solver for the cardiac Bidomain model with a usual coupled solver. The Bidomain model describes the bioelectric activity of the cardiac tissue and consists of a system of a non-linear parabolic reaction-diffusion partial differential equation (PDE) and an elliptic linear PDE. This system models at macroscopic level the evolution of the transmembrane and extracellular electric potentials of the anisotropic cardiac tissue. The evolution equation is...
Domain decomposition techniques provide a powerful tool for the numerical approximation of partial differential equations. We focus on mortar finite element methods on non-matching triangulations. In particular, we discuss and analyze dual Lagrange multiplier spaces for lowest order finite elements. These non standard Lagrange multiplier spaces yield optimal discretization schemes and a locally supported basis for the associated constrained mortar spaces. As a consequence, standard efficient iterative...
Domain decomposition techniques provide a powerful tool for the numerical
approximation of partial differential equations.
We focus on mortar finite element methods on non-matching triangulations.
In particular, we discuss and analyze dual Lagrange multiplier spaces
for lowest order finite elements.
These non standard Lagrange multiplier spaces yield optimal discretization
schemes and a locally supported basis for the associated
constrained mortar spaces. As a consequence,
standard efficient iterative...
We consider two static problems which describe the contact between a piezoelectric body and an obstacle, the so-called foundation. The constitutive relation of the material is assumed to be electro-elastic and involves the nonlinear elastic constitutive Hencky's law. In the first problem, the contact is assumed to be frictionless, and the foundation is nonconductive, while in the second it is supposed to be frictional, and the foundation is electrically conductive. The contact is modeled with the...
We present a domain decomposition theory on an interface problem for the linear transport equation between a diffusive and a non-diffusive region. To leading order, i.e. up to an error of the order of the mean free path in the diffusive region, the solution in the non-diffusive region is independent of the density in the diffusive region. However, the diffusive and the non-diffusive regions are coupled at the interface at the next order of approximation. In particular, our algorithm avoids iterating...
We present a domain decomposition theory on an interface problem
for the linear transport equation between a diffusive and a non-diffusive region.
To leading order, i.e. up to an error of the order of the mean free path in the
diffusive region, the solution in the non-diffusive region is independent of the
density in the diffusive region. However, the diffusive and the non-diffusive regions
are coupled at the interface at the next order of approximation. In particular, our
algorithm avoids iterating...
In this note, we propose and analyse a method for handling interfaces between non-matching grids based on an approach suggested by Nitsche (1971) for the approximation of Dirichlet boundary conditions. The exposition is limited to self-adjoint elliptic problems, using Poisson’s equation as a model. A priori and a posteriori error estimates are given. Some numerical results are included.
In this note, we propose and analyse a method for handling
interfaces between non-matching grids based on an approach
suggested by Nitsche (1971) for the approximation of
Dirichlet boundary conditions. The exposition is limited to
self-adjoint elliptic problems, using Poisson's equation as a
model. A priori and a posteriori error estimates are given. Some
numerical results are included.
A full multigrid finite element method is proposed for semilinear elliptic equations. The main idea is to transform the solution of the semilinear problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and semilinear problems on a very low dimensional space. The linearized boundary value problems are solved by some multigrid iterations. Besides the multigrid iteration, all other efficient numerical methods can also serve as...
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