884 research outputs found
A Nitsche Finite Element Approach for Elliptic Problems with Discontinuous Dirichlet Boundary Conditions
We present a numerical approximation method for linear diffusion-reaction
problems with possibly discontinuous Dirichlet boundary conditions. The
solution of such problems can be represented as a linear combination of
explicitly known singular functions as well as of an -regular part. The
latter part is expressed in terms of an elliptic problem with regularized
Dirichlet boundary conditions, and can be approximated by means of a Nitsche
finite element approach. The discrete solution of the original problem is then
defined by adding the singular part of the exact solution to the Nitsche
approximation. In this way, the discrete solution can be shown to converge of
second order with respect to the mesh size
Incompatibility of time-dependent Bogoliubov--de-Gennes and Ginzburg--Landau equations
We study the time-dependent Bogoliubov--de-Gennes equations for generic
translation-invariant fermionic many-body systems. For initial states that are
close to thermal equilibrium states at temperatures near the critical
temperature, we show that the magnitude of the order parameter stays
approximately constant in time and, in particular, does not follow a
time-dependent Ginzburg--Landau equation, which is often employed as a
phenomenological description and predicts a decay of the order parameter in
time. The full non-linear structure of the equations is necessary to understand
this behavior.Comment: to appear in LM
A C0 interior penalty method for 4th order PDE's
Fourth order Partial Differential Equations (PDE's) arise in many different physic's fields. As an example, the research group for Mathematical and Computational Modeling at UPC LaCàN is studying flexoelectricity, a very promising field which aims to replace some of the uses of piezoelectric materials, and whose equations involve 4th order derivatives. This work provides a method to solve these 4th order PDE's using the Finite Element Method (FEM) with C0 elements, which provides many advantages with respect to other methods that involve using C1 elements or decoupling the equation. The method is developed over the equations of the deformation of a Kirchoff plate, which is also a 4th order PDE. This method is then successfully validated with numerical experiments, both physical and artificial. An analysis of the convergence as well as the method's sensitivity to a newly added parameter is also provided. Due to the success of the method, LaCàN group will use this method to solve flexoelectricity's PDE's
Nitsche's method for a Robin boundary value problem in a smooth domain
We prove several optimal-order error estimates for the finite element method
applied to an inhomogeneous Robin boundary value problem for the Poisson
equation defined in a smooth bounded domain in , . The
boundary condition is imposed weakly by the Nische's method. We also study the
symmetric interior penalty discontinuous Galerkin method and prove the same
error estimates. Numerical examples to confirmed our results are also reported.Comment: 16 pages, 7 figure
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