Approximations for weakly nonlinear evolution equations
Milan Miklavčič · Mathematics of Computation · 1989
Convergence of approximations for a large class of weakly nonlinear parabolic and hyperbolic equations is proven. The main emphasis is on proving convergence of finite element and spectral Galerkin approximations of solutions to the weakly nonlinear wave equation \[ u ( t ) + A u ( t ) = F ( t , u ( t ) , u ′ ( t ) ) , u ( 0 ) = x 0 , u ′ ( 0 ) = y 0 , u(t) + Au(t) = F(t,u(t),u’(t)),\quad u(0) = {x_0},u’(0) = {y_0}, \] under minimal assumptions on the linear operator A and on the approximation spaces. A can be a very general elliptic operator (not just of 2nd order and not necessarily in a bounded domain); A can also be very singular and degenerate. The results apply also to systems of equations. Verification of the hypotheses is completely elementary for a large class of problems.