Galerkin’s Method for Some Highly Nonlinear Problems

Galerkin’s method is analyzed for mixed initial value-boundary value problems for the following two equations: \[ \frac{{\partial u}} {{\partial t}} - \sum\limits_{i = 1}^n {\frac{\partial } {{\partial x_i }}} A_i (x,\nabla u) = f(x,t,u,\nabla u) \] and \[ \frac{{\partial ^2 u}} {{\partial t^2 }} - \sum _{i = 1}^n {\frac{\partial } {{\partial x_i }}} A_i (x,\nabla u) = f(x,t,u,\nabla u). \] Optimal order $H^1 $ and $L^2 $ convergence estimates are obtained.

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