177 research outputs found
Optimal Error Estimates of Galerkin Finite Element Methods for Stochastic Partial Differential Equations with Multiplicative Noise
We consider Galerkin finite element methods for semilinear stochastic partial
differential equations (SPDEs) with multiplicative noise and Lipschitz
continuous nonlinearities. We analyze the strong error of convergence for
spatially semidiscrete approximations as well as a spatio-temporal
discretization which is based on a linear implicit Euler-Maruyama method. In
both cases we obtain optimal error estimates.
The proofs are based on sharp integral versions of well-known error estimates
for the corresponding deterministic linear homogeneous equation together with
optimal regularity results for the mild solution of the SPDE. The results hold
for different Galerkin methods such as the standard finite element method or
spectral Galerkin approximations.Comment: 30 page
A randomized and fully discrete Galerkin finite element method for semilinear stochastic evolution equations
In this paper the numerical solution of non-autonomous semilinear stochastic
evolution equations driven by an additive Wiener noise is investigated. We
introduce a novel fully discrete numerical approximation that combines a
standard Galerkin finite element method with a randomized Runge-Kutta scheme.
Convergence of the method to the mild solution is proven with respect to the
-norm, . We obtain the same temporal order of
convergence as for Milstein-Galerkin finite element methods but without
imposing any differentiability condition on the nonlinearity. The results are
extended to also incorporate a spectral approximation of the driving Wiener
process. An application to a stochastic partial differential equation is
discussed and illustrated through a numerical experiment.Comment: 31 pages, 1 figur
On a randomized backward Euler method for nonlinear evolution equations with time-irregular coefficients
In this paper we introduce a randomized version of the backward Euler method,
that is applicable to stiff ordinary differential equations and nonlinear
evolution equations with time-irregular coefficients. In the finite-dimensional
case, we consider Carath\'eodory type functions satisfying a one-sided
Lipschitz condition. After investigating the well-posedness and the stability
properties of the randomized scheme, we prove the convergence to the exact
solution with a rate of in the root-mean-square norm assuming only that
the coefficient function is square integrable with respect to the temporal
parameter.
These results are then extended to the numerical solution of
infinite-dimensional evolution equations under monotonicity and Lipschitz
conditions. Here we consider a combination of the randomized backward Euler
scheme with a Galerkin finite element method. We obtain error estimates that
correspond to the regularity of the exact solution. The practicability of the
randomized scheme is also illustrated through several numerical experiments.Comment: 37 pages, 3 figure
Optimal Regularity for Semilinear Stochastic Partial Differential Equations with Multiplicative Noise
This paper deals with the spatial and temporal regularity of the unique
Hilbert space valued mild solution to a semilinear stochastic partial
differential equation with nonlinear terms that satisfy global Lipschitz
conditions. It is shown that the mild solution has the same optimal regularity
properties as the stochastic convolution. The proof is elementary and makes use
of existing results on the regularity of the solution, in particular, the
H\"older continuity with a non-optimal exponent
Duality in refined Sobolev-Malliavin spaces and weak approximations of SPDE
We introduce a new family of refined Sobolev-Malliavin spaces that capture
the integrability in time of the Malliavin derivative. We consider duality in
these spaces and derive a Burkholder type inequality in a dual norm. The theory
we develop allows us to prove weak convergence with essentially optimal rate
for numerical approximations in space and time of semilinear parabolic
stochastic evolution equations driven by Gaussian additive noise. In
particular, we combine a standard Galerkin finite element method with backward
Euler timestepping. The method of proof does not rely on the use of the
Kolmogorov equation or the It\={o} formula and is therefore non-Markovian in
nature. Test functions satisfying polynomial growth and mild smoothness
assumptions are allowed, meaning in particular that we prove convergence of
arbitrary moments with essentially optimal rate.Comment: 32 page
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