A convergence result is given for discrete descent based on Sobolev gradients arising from differential equations which may be expressed as quadratic forms. The argument is an extension of the result of David G. Luenberger on Euclidean descent and compliments the work of John W. Neuberger on Sobolev descent.
"A convergence result for discreet steepest decent in weighted sobolev spaces." Abstr. Appl. Anal. 2 (1-2) 67 - 72, 1997. https://doi.org/10.1155/S1085337597000274