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Keywords:
nonlinear boundary value problem; finite elements; rate of convergence; anisotropic heat conduction
nonlinear boundary value problem; finite elements; rate of convergence; anisotropic heat conduction
Summary:
A nonlinear elliptic partial differential equation with homogeneous Dirichlet boundary conditions is examined. The problem describes for instance a stationary heat conduction in nonlinear inhomogeneous and anisotropic media. For finite elements of degree $k\ge 1$ we prove the optimal rates of convergence $\mathcal O(h^k)$ in the $H^1$-norm and $\mathcal O(h^{k+1})$ in the $L^2$-norm provided the true solution is sufficiently smooth. Considerations are restricted to domains with polyhedral boundaries. Numerical integration is not taken into account.
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