BEM Numerical Implementation of the 2D Laplace Equation

Consider the two-dimensional Laplace equation for temperature in the domain presented in 

Since with BEM we are required to apply both boundary conditions, the Dirichlet and Neumann boundary conditions, in the BEM literature they are not referred to as essential and natural.  

By using Green s identities and Green s functions, an equivalent integral equation is obtained as 

If we use constant elements (one node), the value of the temperature and heat are considered constant and equal to the value at the mid point of the element. Therefore, we can place the values for temperature and heat flow outside of the integrals in eqn. (10.21) to get 

we change the element superscript j by a subscript j because, for the constant elements, each node actually represents an element. This equation can be written in compact form as 

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