INTRODUCTION
Let, Ω⊂R^{n} be a bounded domain with the smooth boundary and t>0.
The following initialboundary value problem (Showalter,
1996):
Where:
and b_{ij}(x) = b_{ji}(x), a_{ij}(x) = a_{ji}(x)
(i,j = 1,2,...,n) are continuous functions in and
the following inequality is valid:
For ∀x ∈ ,
∀ξ ∈ ú^{n} was investigated by Petrosyan
and Hakobyan (2008).
The problem in Eq. 13 was investigated
in the case in which L is linear, M is nonlinear where, L and M are degenerated
operators (Petrosyan and Hakobyan, 2008). The solution
of a general case in which L and M are nonlinear is considered by Gaevskii
et al. (1978).
Quarteroni et al. (2000) proved (using Galerkin’s
method) that the solution of the problem 13 exists (Mamikonyan,
2006).
In this study, we construct an approximate solution of the problem 13 using finiteelement method for the case in which Ω∈(0,1)x(0,1)⊂ú^{2}, Lu = Δu,
Definition: The function:
may be a weak solution of the problem in Eq. 13
if:
and for:
the Eq. 4 is valid:
Petrosyan and Hakobyan (2008) proved that the Eq.
4 has a unique solution.
Let, we construct an approximate solution for the problem in Eq. 4 using the finiteelement method.
Suppose the partition domain Ω = (0,1)^{2} with a uniform triangulation of mesh size h with respect to x and y as Fig. 1.

Fig. 1:  Partition
domain Ω = (0,1)^{2} with a uniform triangulation of mesh
size h with respect to x and y 
x_{i+1} x_{i} = h,i,j = 1,2,...,n,
y_{j+1}y_{j} = hh = 1/n

We construct the piecewise linear functions φ_{ij}(x,y) by the
following rule:
and linear in the domain of every triangle. In the remained triangles of the
square [0,1]x[0,1] we assume φ_{ij} (x,y) = 0.
In continue, we set N = (n1)^{2} as the basis functions. Let, ω_{n} = {(ih, jh); i, j = 1,2,...,n1}. If we number the points of the set ω_{n} for example:
then, the basis functions φ_{ij} will be renumbered, i.e., by
constructing ψ_{k}(A_{r}) δ_{kr} (k,r = 1,2,...,n1)
we get the system ψ_{1},ψ_{2},...,ψ_{n}.
Note that S_{n} is the linear space generated by the functions ψ_{i}
= (i = 1,2,...,N) and dimS_{n} = N and where,
v is linear in every triangle and v = 0 on ∂Ω}.
It is easy to see that:
is a subspace. To calculate,
we use the Table 1.
Table 1:  The
derivatives of the basis functions ψ_{i} 

Denote:
To find the weak solution of the problem 13 we use the Galerkins method:
which is equivalent to:
We can rewrite the Eq. 5 in the matrix form:
where, β_{N} = (α_{1},α_{2},...,α_{N}),
It is easy to check that the matrix M_{N} has the following form:
where, E is the unit matrix and A is the following matrix of order of (n1)x(n1):
The solution of the system of differential Eq. 6 with the
conditions:
where, c_{i} are the coefficients of the expansion of the function u_{0}(x,y) by the basis ψ_{i}(x,y) (i =1,...,N) may be denoted by using α_{i}(t) (i = 1,...,). Thus, we obtain the following sequence of the functions:
This sequence:
converges in:
norm to the weak solution of the problem in Eq. 13.
To find the numerical solution of the system 6 we use the θ method (Braess,
2001). Suppose the partition [0,T] into equal parts with the step Δt
denote by:
Now we replace the system 6 by the following different system:
where,
For every k, we get the linear system of equations.
We choose the parameter θ such that the matrix:
will be positive,
Then we can represent the system of Eq. 8 in the following
form:
where, K = H^{T}H.
Denote by:
It is easy to verify that:
CONCLUSION
The initial boundary value problem is investigated for the pseudoparabolic equation with nonlinear operators. An approximate solution for this problem is obtained using the finite element method. Finally it is proved that the constructed sequence converges to the exact solution is possible.