Existence and Finite Element Approximation of Nonlinear Elliptic Partial Differential Equations

Abstract

A nonlinear elliptic partial differential equation is studied in this paper to determine the existence and finite element approximation of a class of elliptic partial differential equations, which are defined on a bounded domain with the appropriate boundary conditions. The variational formulation of the problem is developed, in which the concept of weak solution is utilized in order to handle the nonlinearities and lack of regularity of the data. The solution is proven to exist and under appropriate conditions, it is shown to be unique, with standard functional analysis tools, such as monotonicity and compactness arguments. A finite element method that builds the conforming piecewise polynomial spaces is developed to find the approximate solution numerically. This discrete problem is then analyzed so that there is consistency and stability. The a priori estimates of errors are obtained in standard energy and square-integrable norms and they show that the numerical solution approaches the accurate solution as the mesh is refined. Theoretical convergence rates are set in place and proven to be based on the regularity of the exact solution and the degree of the finite element space. The theoretical results are verified using numerical experiments. Benchmark problems that have known analytical answers are taken into consideration, and the errors are calculated as mesh refinements. The obtained results validate the convergence rates predicted and demonstrate the validity and correctness of the developed method. On the whole, the research offers a coherent approach that involves a combination of stringent existence theory and credible finite element representation and computational verification.