A priori \(L^2\) error analysis for an expanded mixed finite element method for quasilinear pseudo-parabolic equations
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 \(d\).
scientific article; zbMATH DE number 6248006