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Many engineering problems are expressed as differential or integral equations with proper boundary and initial conditions. Mostly, these equations are not solvable analytically. It is for this reason approximate numerical solutions are attempted.

Boundary element method is one of the numerical methods that is employed for solution of such equations. It can be effectively used for the solution of 2 and 3D linear equations that arise from many engineering problems.

You may find a short presentation on the advantages and disadvantages of the boundary element method in:
Presentation on BEE 1.0

You may find simplified lecture notes on boundary element method in:
Simplified lecture notes on boundary element method