"Mathematically mature" book on numerical analysis of PDE

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I'm looking for a mathematically mature textbook on numerical analysis of PDE with a focus toward theory. Every book I look at seems to state in the preface that the book makes mathematical concession for readability for engineers, physicists, and the like. I'm looking for nice, general results (with rigorous proof) about convergence of finite difference schemes and on similar issues. Ideally, the book also shouldn't focus on the special case of two independent variables and should include a treatment nonlinear PDE (though perhaps there isn't much general theory for the nonlinear cases, so this isn't necessary).