Spectral methods, or any methods where you have chosen a basis of continuous functions and are solving for weights produces solutions in the continuous domain. That's not a discrete model.
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The continuous models are an ad-hoc, purely mental, construction. When you have to solve a PDE, you actually build a discrete model (using finite elements), and solve the discrete thing. Except in very simple toy problems, you can never "solve" anything using only continuous tools.