A priori \(L^2\) error analysis for an expanded mixed finite element method for quasilinear pseudo-parabolic equations (Q2870475)

The authors consider a quasilinear pseudo-parabolic equation (in which the flux contains nonlinearities and the time derivative of the gradient of the solution) in a convex \(d\)-dimensional domain (\(1\leq d\leq 3\)) along with second-kind boundary and initial data in Sobolev spaces \(W^{s,p}\) for the space variables at \(L^2\) spaces in time. Assuming smooth nonlinearities and a Lipschitz-continuous right hand side, they use a regular quasi-uniform grid to discretize the space variables by finite elements. Solving for the solution, its gradient and the flux, they need 3 different functions and discretization spaces and a corresponding expanded mixed finite element weak formulation. For this, they prove existence of the semidiscrete solution by reformulating the task as a fixed point problem and using the Brouwer existence theorem. Further, they prove the convergence of the semidiscrete solution being optimal with respect to smoothness, polynomial degree and dimension \(d\).