| Summary: | We consider the analysis of a one-parameter family of $hp$--version discontinuous Galerkin finite element methods for the numerical solution of quasilinear parabolic equations of the form $u'-\na\cdot\set{a(x,t,\abs{\na u})\na u}=f(x,t,u)$ on a bounded open set $\om\in\re^d$, subject to mixed Dirichlet and Neumann boundary conditions on $\pr\om$. It is assumed that $a$ is a real--valued function which is Lipschitz-continuous and uniformly monotonic in its last argument, and $f$ is a real-valued function which is locally Lipschitz-continuous and satisfies a suitable growth condition in its last argument; both functions are measurable in the first and second arguments. For quasi--uniform $hp$--meshes, if $u\in \H^1(0,T;\H^k(\om))\cap\L^\infty(0,T;\H^1(\om))$ with $k\geq 3\frac{1}{2}$, for discontinuous piecewise polynomials of degree not less than 1, the approximation error, measured in the broken $H^1$ norm, is proved to be the same as in the linear case: $\mathscr{O}(h^{s-1}/p^{k-3/2})$ with $1\leq s\leq\min\set{p+1,k}$.
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