We construct a theory of general boundary value problems for differential operators whose symbols do not necessarily satisfy the Atiyah-Bott condition [3] of vanishing of the corresponding obstruction. A condition for these problems to be Fredholm is introduced and the corresponding finiteness theorems are proved.

Many partial differential equations (PDEs) admit conserved quantities like mass or energy. Those quantities are often essential to establish well-posed results. When approximating a PDE by a finite-difference scheme, it is natural to ask whether related discretized quantities remain conserved under the scheme. Such conservation may establish the stability of the numerical scheme. We present an algorithm for checking the preservation of a polynomial quantity under a polynomial finite-difference scheme. In our algorithm, schemes can be explicit or implicit, have higher-order time and space derivatives, and an arbitrary number of variables. Additionally, we present an algorithm for, given a scheme, finding conserved quantities. We illustrate our algorithm by studying several finite-difference schemes.