JP1.15 THE CONSTRUCTION OF THE NUMERICAL SCHEMES IN THE HERMITIAN FINITE ELEMENTS SPACES Ireneusz A. Winnicki * Military University of Technology (MUT), Warsaw, Poland 1. INTRODUCTION A number of different methods have been devel- oped for numerical modeling of initial-boundary value problems of hydrodynamics. The most popular ones are: the finite differences, spectral, pseudo-spectral and collocation methods. The finite element methods have been known for over thirty years. Their theoretical dis- cussion was presented by Zienkiewicz (1971) and Strang and Fix (1973). Cullen (1974) was the first to use the finite ele- ments in modeling of atmospheric processes. His ex- ample was followed by Staniforth and Daley (1977) and Staniforth and Mitchell (1977). In their considerations they mainly applied Lagrange’s approximations, which for any variable v can be written in the form Σv i χ i , where χ i , basis functions, usually linear or low-order polynomials, v i – the value of v at the node i. On the basis of the theory of approximation Strang and Fix (1973) shortly described a second approach to the problem of solving partial differential equations using finite elements. They defined new finite elements spaces, called Hermitian spaces, and suggested that unknown solutions could be approximated in the form: ∑ ψ + ϕ = ] [ ' ) ( ) ( ) ( ) ( ) ( x t v x t v x,t v i i i i h (*) Ten years later Rymarz and Winnicki (1984) analyzed more complicated finite element approximations, which were written in the general form: ∑ γ + ψ + ϕ = ] [ ' ' ' ) ( ) ( ) ( ) ( ) ( ) ( ) ( x t v x t v x t v x,t v i i i i i i h (**) where ) ( x,t v h is a discretization of the exact solution ) , ( t x v ; ' ' ' , i i v v - the first and the second derivative of the solution. 2. THE FINITE ELEMENT SPACES We will concentrate on the finite element approxi- mation for initial-boundary value problem:      = ψ + ϕ = > ψ + ϕ = ∑ ∑ 0 , )] ( ) ( ) ( ) ( [ 0 )], ( ) ( ) ( ) ( [ ' , 0 , 0 , 0 ' t x t x t t x t x t i i i i h i i i i h f f f f f f (1) where f i vector of values of sought-for functions f h at the node i; h - mesh spacing; ' i f - values of the first deriva- tives of the solution f ; ) ( ), ( x x i i ψ ϕ - Hermitian space basis functions. It is important to notice that each basis function has the compact support and this support is small. It is given by a few elements of the space. These functions vanish at the nodes outside the element and take the value 1 at the node i. For (*) the assumptions hold piece-wise cubic functions. Now we can define Hermitian space ) 3 ( h V : } , 0 , 0 , 0 , ,..., 1 ), ( ], , 0 [ : { ) ( 3 1 ) 3 ( l x x Dv v N i K P v l C v v V h i K h = = = = = ∈ ∈ = (2) The basis of ) 3 ( h V is a pair of cubic piece-wise polyno- mials. They are of 1 C type. The space ) 3 ( h V is a Hilbert space, ) , 0 ( ~ 2 0 ) 3 ( l H V h ⊂ , endowed by the scalar product: ∫ Ω Ω = dx x v x w v w H ) ( ) ( ) , ( ) ( ~ 2 0 (3) 3. PROBLEM FORMULATION FOR DIFFUSION EQUATION Let us consider the linear one-dimensional diffusion equation with irregular initial condition: ) ( ) 0 , ( , 0 0 2 2 2 x u x u x u a t u = = ∂ ∂ - ∂ ∂ (4) After applying the finite element method’s technique in the Hermitian space ) 3 ( h V for ) , ( t x v u h h = (HFEM):        ψ = ψ ϕ = ϕ = ψ ∂ ∂ - ∂ ∂ = ϕ ∂ ∂ - ∂ ∂ ) ), ( ( ) ), 0 , ( ( ) ), ( ( ) ), 0 , ( ( 0 ) , / / ( 0 ) , / / ( 0 0 2 2 2 2 2 2 k k k k k k x u x u x u x u x u a t u x u a t u (5) we obtain the system of implicit discrete equations:          ψ = ϕ = = β + = β + ∫ ∫ + - + - + + + + 1 1 1 1 ) ( ) ( ) / 420 ( ) ( ) ( ) / 420 ( 0 2 0 ' 0 0 ' 1 ' 2 1 ' 1 2 1 i i i i x x i i x x i i n i n i n i n i n i n i dx x x u h H dx x x u h H H L H H L H (6) ' ' 1 ' 1 1 1 ' ' 1 ' 1 1 1 1 1 ' ' 1 ' 1 ' ' 1 ' 1 1 1 112 ) ( 14 ) ( 42 ) ( 42 ) 2 ( 504 ) ( 13 8 ) ( 3 ) ( 13 312 ) ( 54 i i i i i i i i i i i i i i i i i i i i i i i i hu u u h u u L u u h u u u L u u hu u u h H u u h u u u H + + - - = - - + - - = - - + + - = - + + + = + - + - + - + - + - + - + - + - (7) where β = a 2 τ/h 2 ; τ, h - time integrating and spatial steps; a 2 – diffusion coefficient; a 2 = const > 0. The values ' , i i u u are usually called the nodal parameters of the Hermitian space ) 3 ( h V . System Eqns. (6) is equivalent to the system of lin- ear algebraic equations. We rewrite it in the form: * Corresponding author address Prof. Ireneusz Winnicki, Mil. Univ. of Techn., Dept. of Civil Engineering, Chemistry and Applied Physics, Institute of Meteorology, 00-908 Warsaw, Poland, ul Kaliski 2, +48 22 6839475, irekwin@ack.wat.waw.pl This paper is sponsored by the Committee for Scientific Re- search, Warsaw, Poland; grant No 0 T00A 004 17.