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where <math>\Omega</math> is a connected open region in the <math>(x,y)</math> plane whose boundary <math>\partial \Omega</math> is nice (e.g., a [[smooth manifold]] or a [[polygon]]), and <math>u_{xx}</math> and <math>u_{yy}</math> denote the second derivatives with respect to <math>x</math> and <math>y</math>, respectively.
The problem P1 can be solved directly by computing [[antiderivative]]s. However, this method of solving the [[boundary value problem]] (BVP) works only when there is one spatial dimension. It does not generalize to higher-dimensional problems or problems like <math>u+
Our explanation will proceed in two steps, which mirror two essential steps one must take to solve a boundary value problem (BVP) using the FEM.
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