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The Finite Element Methods was used to solve the nonlinear electron plasma equations for the System-Generated Electromagnetic Pulse boundary layer in one spatial dimension. These equations were solved in distance-velocity phase space using a rectangular finite element mesh. Linear approximations were used for both the trial and weight functions for each element. The advection terms in the Vlasov plasma equation were treated with the Heinrich upwinding technique. The time integration was performed using an explicit two-step Lax-Wendroff procedure. The system of algebraic equations were solved with a fully-packed Gauss-Seidel iteration scheme. A value of 2/3 for the upwinding parameter was found to provide the best compromise between dispersion of the pulse, and computer storage requirements. The savings in computer memory results in increased execution speed for the algorithm. Also, it is shown that the numerical scheme does not permit spurious pulse reflections from the edges of the mesh. Results for several test cases are presented. Comparisons are given which show favorable agreement for the finite element technique with other solution relationships.