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Reliable design is dependent on reliable engineering simulation. Many problems of engineering analysis rely on numerical techniques for solving partial differential equations. The foremost method of obtaining approximate solution to partial differential equations is the finite element method. However, alternate approaches, such as the boundary element method, have been proven to be more accurate in certain types of problems. In this work, a new method is developed to quantify the discretization error in boundary element analysis. The method considers the discretization error as an interval variable and a scheme is developed to obtain sharp results of the interval guaranteed to bound the solution. Thus, the error bounds are in terms of worst case bounds that can be used directly in design decisions. Exemplars are presented showing that the obtained interval solutions enclose the closed form solutions.