CANADA'S SPECIAL elections this week have done serious damage to Prime Minister Pierre Elliott Trudeau, as he faces national elections next spring. Canadian politics will now proceed on two levels. While his Liberal Party prepares itself in the conventional ways for the elections, there will be a good deal of pulling and hauling internally over his leadership. Mr. Trudeau has grimly declared that he intends to remain at the head of the party. But that may not necessarily be the last word.

The special elections were for 15 parliamentary seats, seven held by Liberals. Of the 15, the Liberals won two. Ontario, with a third of Canada's population, is its political center of gravity, and five of the previously Liberal seats were there. The Liberal lost all of them and, worse, got fewer votes than either of the two principal opposition parties.

The Liberals' opposition is split between one party to their right and another to their left. A good many Liberals suspect that the defeats this week were owed to votes specifically against Mr. Trudeau. It would follow that, if he retired, the party might hope to win once again by virtue of its strong centrist position. But Mr. Trudeau is not the retiring sort.

He is now in his eleventh year as prime minister. He's been in power longer than the head of any other government among the world's major democracies. A man of brilliance and style, he has devoted himself above all to the cause of national unity. He came to office as the man who, more than any other, might be able to hold the French-speaking and English-speaking populations together. A decade later, the separatist movement now seems in fact to be losing momentum. The separatists who control the Quebec provincial government get more and more vague about the promised referendum on Quebec independence, and more and more, defensive about Quebec's disproportionately severe economic troubles. Stagflation is simultaneously undermining both Mr. Trudeau's position and that of his separatist enemies.

The inflation rate in Canada is a title lower than in the United States. But unemployment, at 8.5 percent last month, is much higher. The Canadian dollar has been sinking on the international exchanges even faster than its American cousin. There seems to be a current of opinion in the English-speaking provinces that Quebec separatism, Mr. Trudeau's special preoccupation, is a diminishing threat - while the state of the economy is a rising one. New circumstances sometimes require new politicans. The special elections have forced Canada to consider the thought, as the long winter arrives.