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Suppose if we take $X_h(G)$ as finite element space then this space (space of piecewise constant basis function)is defined as

$$X_h=\{v: v|_{T}=c_{T}, T \in \mathbb{T}\},$$ where $\mathbb{T}$ is a subdivision of domain $G$. Then my main question is

"How to define (or fix value) of constant $c_T$"?

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    $\begingroup$ I'm not sure if I understand your question. The $c_T$'s are your unknown degrees-of-freedom. They are free to vary -- that's what makes $X_h$ a space, not just a function. A member of that space, say $u \in X_h$, would correspond to defining the values of the $c_T$'s. $\endgroup$ – LedHead Aug 23 at 17:49
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A function in your space is defined as piecewise constant, hence any function living in this space assumes a constant value for each of the elements (the function value if evaluated in the corresponding element).

Your definition may look confusing at first, but there really is no need to fix any particular element-dependent value for implementation. Just take 1 as your local shape function and treat the integration and assembly procedure in the usual way. The coefficients you are asking about turn out to be your unknown degrees of freedom when setting up the discrete system of equations.

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