Fully Discrete A-? Finite Element Method For Nonlinear Electromagnetic Equations

This article is devoted to the study of a fully discrete A—? finite element method to solve the nonlinear electromagnetic equations based on backward Euler dis-cretization in time and nodal finite elements in space. Here we use a power-law for-m, ??|E|? = |E|^?-1,0 < ? < 1, as the nonlinear term, this function is monotonic and? —H???lder continuous property. We design a nonlinear time-discrete scheme for approxi-mation in H₀¹???×H₀¹??? spaces. Then we prove the existence and uniqueness of these discretized fields using the theory of monotone operators. Afterwards we define some in-terpolations of the discretized fields in time, that is the equivalent form, then we prove the solution of the equivalent form converge to the weakness solution of the original problem as ??0. Under the low regularity requirement, then we get an error estimate with O(?^1/2)for time discretization. We use the similar methods to handle the fully discretization, and we get the convergence. Under the lower regularity assumption we get an error estimate with O(?^1/2 + h^1/2). At the end of this paper, under a higher regularity assumption we improved the results of error estimation to O(?+h^min{1,?}).Finally, we can support the theoretical result by some numerical experiments.