Reduced mixed finite element method with a priori and a posteriori error analysis

This thesis presents a novel finite element method (FEM) for the accurate and efficient approximation of the flux variable in second-order elliptic boundary value problems. Traditional mixed FEMs used to solve these problems approximate both the flux and primary variables simultaneously by solving a coupled linear system of algebraic equations (LSAEs). However, in many applications, the flux is the main quantity of interest, and computing the primary variable is unnecessary. We introduce a new finite element scheme that directly approximates the flux without computing the primary function. With the elimination of the primary variable, the LSAEs in our method are positive definite and involve a significantly smaller number of degrees of freedom than the indefinite system in the mixed method. Moreover, any conforming finite element space can be used for flux approximation, whereas the mixed method requires a pair of approximation spaces that satisfy the discrete inf-sup stability condition, necessitating careful consideration. The thesis establishes four main results. First, we develop a new FEM that approximates the flux independently of the primary variable. This is achieved by formulating a variational equation in the flux, eliminating the primary variable through the use of a user-defined parameter δ ∈ (0, 1]. Next, we build upon the proposed FEM by iterating the variational equation on a fixed mesh, incorporating the computed flux into the right-hand side data. This improvisation results in a considerably better flux approximation. The accuracy improvement is more pronounced for smaller values of the parameter δ ∈ (0, 1]. An analysis of the role of δ and the effect of iteration in improving accuracy is presented. Third, we develop an efficient adaptive finite element method motivated by the proposed iterative scheme, where the solution from a coarser mesh is incorporated into the right-hand side terms when solving the variational equation on a finer mesh. Finally, we establish a fully computable error bound for the computed flux. Numerical experiments are provided to confirm the theoretical findings.