Galerkin's method for equations with a small parameter in the highest order derivatives

The question considered is the convergence of Galerkin's method for the operator equation $A_\varepsilon u-Ku\equiv\varepsilon A_1u+A_0u-Ku=f$, where $A_0$ is positive definite, $A_1$ is positive semidefinite with domain of definition $D(A_1)\subset D(A_0)$, and $\varepsilon>0$ is a small parameter. With certain additional natural assumptions it is shown that the solution obtained by Galerkin's method is uniformly convergent to the true solution in the norm defined by the quadratic form $(A_\varepsilon u,u)$ for $0\leqslant\varepsilon\leqslant1$.
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