Finite Volume Element Method For Nonlinear Equations

The research of nonlinear elliptic and parabolic equations are of great importance not only in theory but also in application.In this article,we studies finite volume element method for nonlinear elliptic and parabolic equations.For the nonlinear elliptic equation,a quadratic finite volume method on the triangular meshes is proposed.We prove the boundedness and ellipticity of the bilinear form and obtain the optimal error estimates not only in H1-norm but also in L2-norm.Optimal dual partition firstly proposed in linear equations is adopted.In the error estimate,the dependency between the error in H1-norm but also in L2-norm is used.In addition,we consider the effect of numerical integration formula on the accuracy of high-order schemes.Under certain integration formulas,we obtain optimal error estimate in H1-norm and L2-norm.To confirm the theoretical analysis,we solve the nonlinear equation by the Newton iteration method and prove the quadratic rate of convergence.The numerical results show the effectiveness of our method.For the parabolic equation with nonlinear time-derivative term and nonlinear diffusion term,a linear finite volume element is introduced.For the fully discrete scheme,we prove the optimal error estimates in H1-norm and L2-norm.For the nonlinear time derivative,we use the result in finite element method and the triangular inequality to accomplish the proof.For this nonlinear equation,we devise a nonlinear iteration scheme to solve it.The numerical results show the correctness of our method.In addition,according to the relationship of temperature and energy,the radiation diffusion equations can be transformed into a strongly nonlinear equation with nonlinear time derivative and nonlinear diffusion term.For this equation,we construct two finite volume element schemes: conservative scheme and non-conservative scheme.In the conservative scheme,the whole nonlinear time derivative is differentiated.In the nonconservative scheme,the variable is differentiated.The numerical results show that the conservative scheme can keep the energy constat and the non-conservative scheme violates the law of energy conservation.