Mathematical Physics - Volume II - Numerical Methods

3.6 Program for solving of elliptical problems

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• y coordinate of the first node (float) • x coordinate of the last node (float) • y coordinate of the last node (float 3. A data group contains data based on which series of elements can be generated, by using data for previously generated nodes. • Number of elements in a series (integer) • Difference between serial numbers of corresponding nodes which belong to adjacent elements in the series (integer) • Number of nodes in each element (6 or 9) (integer) • Serial number of subregion which the series elements belong to (integer) • 6 or 9 serial numbers (integers) of the first element in the series (pay attention the standard distribution of nodes in the element). 4. grupa su podaci o vrednostima konstanti k , b i ¯ f • Values of constant k (float) • Values of constant b (float) • Values of constant ¯ f (float) 5. grupa su podaci o eventualnim singularitetima funkcije f . • Number of nodes which contain singularities (integer) • Value of ˆ f (float). 6. A data group contains essential boundary conditions. The same value of function ˆ u can be defined along the entire line of nodes. • Serial number of the first element in the series (integer) • Number of nodes in the series (integer) • Difference between adjacent nodes in the series (integer) • Values of ˆ u (float). 7. A data group defines natural boundary conditions. These conditions are also possible to define on the entire series of elements along the boundary. • Serial number of the first element in the series (integer) • Number of elements in the series (integer) • Difference in serial numbers of adjacent elements (integer) • Serial number of the side located at the region boundary (pay attention to the standard numbering of sides within an element) (integer)

• Value of p (float) • Value of γ (float).

3.6.3 Examples

∂ 2 u ∂ x 2

∂ 2 u ∂ y 2

1. Example. We will solve the equation = 4 for a 2 × 2 square. All boundary conditions will be natural, except in ( 0 , 0 ) , wherein the unknown function u is equal to zero. Natural boundary conditions are chosen so that the solution of the problem is u = x 2 + y 2 . Input database is as follows: +

----------------------------------------------------------- Test primer br. 1: Jednacina Lap(u)=4. Resenje: u=x^2+y^2. ----------------------------------------------------------- 3

1 3 1 0. 0. 2. 0. 4 3 1 0. 1. 2. 1. 7 3 1 0. 2. 2. 2.