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Multiplying (A.1) and (A.2) through by test functions and integrating by parts yields (h, a) / - (#h,V - a)T + (Trh, a),T h, a) T -(V KEh T n,"-),Th - h, - a -n OT = 0, Th, T (Tq~h, P) a~h + )~h= (A.4) (-pco, a)T,, (A.5) (g, w' )Th~ (A.6) Note that the numerical flux given by equation (A.3) has been substituted into the system. This is possible since the problem is linear. The resulting elemental matrix system has the form AK -BK BT DK CK EK -- CK QK -EK UK MK 0 (A7) =FK JK GK The upper left quadrant is again block diagonal and may be inverted to reduce the system to degrees of freedom on the boundaries. Note that the system in this case has fewer unique sub-matrices. This facilitates matrix assembly by reducing required computations. The final form of the global problem is HO = R. (A. 8) Newton iterations are not required since the problem is linear. The accuracy of the solution is then limited by the mesh and element order. 80