1,016 research outputs found
Finite element methods for semilinear elliptic stochastic partial differential equations
We study finite element methods for semilinear stochastic partial differential equations. Error estimates are established. Numerical examples are also presented to examine our theoretical results
Adaptive Pseudo-Transient-Continuation-Galerkin Methods for Semilinear Elliptic Partial Differential Equations
In this paper we investigate the application of pseudo-transient-continuation
(PTC) schemes for the numerical solution of semilinear elliptic partial
differential equations, with possible singular perturbations. We will outline a
residual reduction analysis within the framework of general Hilbert spaces,
and, subsequently, employ the PTC-methodology in the context of finite element
discretizations of semilinear boundary value problems. Our approach combines
both a prediction-type PTC-method (for infinite dimensional problems) and an
adaptive finite element discretization (based on a robust a posteriori residual
analysis), thereby leading to a fully adaptive PTC-Galerkin scheme. Numerical
experiments underline the robustness and reliability of the proposed approach
for different examples.Comment: arXiv admin note: text overlap with arXiv:1408.522
Weak order for the discretization of the stochastic heat equation
In this paper we study the approximation of the distribution of
Hilbert--valued stochastic process solution of a linear parabolic stochastic
partial differential equation written in an abstract form as driven by a Gaussian
space time noise whose covariance operator is given. We assume that
is a finite trace operator for some and that is
bounded from into for some . It is not required
to be nuclear or to commute with . The discretization is achieved thanks to
finite element methods in space (parameter ) and implicit Euler schemes in
time (parameter ). We define a discrete solution and for
suitable functions defined on , we show that |\E \phi(X^N_h) - \E
\phi(X_T) | = O(h^{2\gamma} + \Delta t^\gamma) \noindent where . Let us note that as in the finite dimensional case the rate of
convergence is twice the one for pathwise approximations
Branching diffusion representation of semi-linear elliptic PDEs and estimation using Monte Carlo method
We study semi-linear elliptic PDEs with polynomial non-linearity and provide
a probabilistic representation of their solution using branching diffusion
processes. When the non-linearity involves the unknown function but not its
derivatives, we extend previous results in the literature by showing that our
probabilistic representation provides a solution to the PDE without assuming
its existence. In the general case, we derive a new representation of the
solution by using marked branching diffusion processes and automatic
differentiation formulas to account for the non-linear gradient term. In both
cases, we develop new theoretical tools to provide explicit sufficient
conditions under which our probabilistic representations hold. As an
application, we consider several examples including multi-dimensional
semi-linear elliptic PDEs and estimate their solution by using the Monte Carlo
method
A modified semi--implict Euler-Maruyama Scheme for finite element discretization of SPDEs with additive noise
We consider the numerical approximation of a general second order
semi--linear parabolic stochastic partial differential equation (SPDE) driven
by additive space-time noise. We introduce a new modified scheme using a linear
functional of the noise with a semi--implicit Euler--Maruyama method in time
and in space we analyse a finite element method (although extension to finite
differences or finite volumes would be possible). We prove convergence in the
root mean square norm for a diffusion reaction equation and diffusion
advection reaction equation. We present numerical results for a linear reaction
diffusion equation in two dimensions as well as a nonlinear example of
two-dimensional stochastic advection diffusion reaction equation. We see from
both the analysis and numerics that the proposed scheme has better convergence
properties than the standard semi--implicit Euler--Maruyama method
- …