General Kvashnin in 1995 (in charge of planning the 2nd Chechen war, Chief of General Staff Russian armed forces):
"We will beat the Chechens to pulp, so that the present generation will be too terrified to fight Russia again. Let Western observers come to Grozny and see what we have done to our own city, so that they shall know what may happen to their towns if they get rough with Russia. But you know, Pavel, in 20–30 years a new generation of Chechens that did not see the Russian army in action will grow up and they will again rebel, so we’ll have to smash them down all over again"
Ukrainian Chief of General staff Zalzuzny also told Economist that the Russian approach is similar to the 2nd Chechen War.
I would say that a major difference is in scale; Ukraine is massive compared to Chechnya. The battle of Grozny is comparable to the battle of Mariupol alone.
My hunch is that Putin/Gerisimov are effected by Chechnya and will manage things in a similar but also different way. They know that they had a 8-9 year insurgency in Chechnya and would want to finish off the Ukrainians in a way that makes the insurgency smaller after the conventional ground war.
Gerisimov also has advocated "hybrid warfare"/asymmetrical war for his career, which is fighting war in a staggered and strange way with emphasis on economic and political factors while abandoning classical approaches to conventional war fighting. I think this is part of the reason why the Russian approach to Ukraine is so bizarre. In order to surprise NATO they had to start the war with a small, underfunded army and now they are scaling up with triple the budget. They are being helped militarily by China, Iran, North Korea plus economic help from China, Turkey, India and other allied countries.
I think they will try to turn Ukraine into the European Somalia by making them ultra-poor. Their GDP per capita has already dropped to around $2000/per person. They will also try to drag out the war to be very long, going for many years and will try to kill and cripple as many Ukrainian fighters and future insurgents as possible.
Ukraine requests 50 billion dollars in 2022 just to keep the country running, plus 2 billion dollars a month to repair the damage from Russian missile strikes to civilian infrastructure. In the short run the West can donate them but as the war goes on for years it is less likely. Also excess NATO equipment and stockpiles are depleted and Ukraine is reliant on new production. All in all this is costing the West hundreds of billions of dollars and if the war continues for years, it will turn into a multi-trillion dollar war for the West.
Now I see that Putin has literally 'outsourced' the attack combat to the private sector with mercenary forces. I think he is doing this because in the Chechen wars, he had to deal with a lot of political problems from the use of Russian conscripts in the war. Instead with PMCs he doesn't have to deal with the political risks and damage to his regime.