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4.3 Stochastic Finite Elements
methods. In the following, we present an approach, where we discretize the SPDE directly. To make
the approach more illustrative we demonstrate the method by using a parabolic SPDE
∂ t u(t,x,ω) − u xx (t,x,ω) = f (t,x,ω) . (4.5)
The temporal and spatial derivatives are determined using well-known approximations. Using the
explicit Euler scheme for the discretization of the time derivative, we get
u(t + τ,x,ω) = u(t,x,ω) + τ(u xx (t,x,ω) + f (t,x,ω)) . (4.6)
Discretizing the spatial derivative using central differences, the fully discrete equation is
( )
u(t,x + h,ω) − 2u(t,x,ω) + u(t,x − h,ω)
u(t + τ,x,ω) = u(t,x,ω) + τ
+ f (t,x,ω)
h 2
. (4.7)
The stochastic quantities in this equation are approximated by using a truncated polynomial chaos
expansion leading to a numerical scheme that needs methods for the addition and multiplication of
polynomial chaos expansions. Section 3.3 presents numerical methods for this task.
The main drawback of these methods is that computations in the polynomial chaos require the
solution of linear systems of equations. Furthermore, the construction of unstructured or adaptive
grids is complicated in comparison to the generation of adaptive grids for finite elements.
The advantage of stochastic finite difference methods is the simple possibility to parallelize explicit
stochastic finite difference schemes, because the computations on different nodes are independent.
4.3 Stochastic Finite Elements
It is well-known that the variational formulation of a deterministic PDE is
find u ∈ V such that a(u,v) = b(v) ∀v ∈ V , (4.8)
where a(·,·) is a bilinear form related to the PDE and b(·) a linear form related to the right hand side
of the PDE. The space V is the space of all admissible functions, e.g. the Sobolev space H0 1 for the
simple prototype equation −∇ · (k∇φ) = f in D, φ = 0 on ∂D.
For stochastic coefficients, right hand sides or boundary conditions, the bilinear and/or linear
form become stochastic quantities. Denote by a(u,v,ω), b(v,ω) the dependence of the forms on the
stochastic event ω ∈ Ω. The aim of the stochastic problem is to find a random field, i.e. an element of
the tensor product space V ⊗S , u ∈ V ⊗S , where S is the space of random functions, e.g. L 2 (Ω),
the space of all random variables with finite second order moments. The weak formulation of the
stochastic problem is:
find u ∈ V ⊗ S such that A(u,v) = B(v) ∀v ∈ V ⊗ S , (4.9)
where
∫
A(u,v) = a(u,v,ω)dω = E(a(u,v,ω)) (4.10)
Ω
and
∫
B(v) = b(v,ω)dω = E(b(v,ω)) . (4.11)
Ω
The weak formulation of an SPDE is simply the expectation of the deterministic problem (4.8).. To
ensure existence and uniqueness of a solution, we need the form A to be continuous and coercive and
the form B to be continuous on the space V ⊗S . Hence, coercivity and continuity are ensured if the
forms a and b are coercive, respectively continuous, for elementary events ω ∈ Ω almost sure and
such that the Wick product is well-defined (cf. Section 3.2.1).
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