Coercivity is a characteristic property of the bilinear term in a weak form of a partial differential equation in both infinite space and the corresponding finite space utilized by a numerical scheme. This concept implies \textit{stability} and \textit{well-posedness} of the weak form in both the exact solution and the numerical solution. In fact, the loss of this property especially in finite dimension cases leads to instability of the numerical scheme. This phenomenon occurs in three major families of problems consisting of advection-diffusion equation with dominant advection term, elastic analysis of very thin beams, and associated plasticity and non-associated plasticity problems. There are two main paths to overcome the loss of coercivity, first manipulating and stabilizing a weak form to ensure that the discrete weak form is coercive, second using an automatically stable method to estimate the solution space such as the Discontinuous Petrov Galerkin (DPG) method in which the optimal test space is attained during the design of the method in such a way that the scheme keeps the coercivity inherently. In this dissertation, A stable numerical method for the aforementioned problems is proposed. A stabilized finite element method for the problem of migration risk problem which belongs to the family of the advection-diffusion problems is designed and thoroughly analyzed. Moreover, DPG method is exploited for a wide range of valuing option problems under the black-Scholes model including vanilla options, American options, Asian options, double knock barrier options where they all belong to family of advection-diffusion problem, and elastic analysis of Timoshenko beam theory. Besides, The problem of American option pricing, migration risk, and plasticity problems can be categorized as a free boundary value problem which has their extra complexity, and optimization theory and variational inequality are the main tools to study these families of the problems. Thus, an overview of the classic definition of variational inequalities and different tools and methods to study analytically and numerically this family of problems is provided and a novel adjoint sensitivity analysis of variational inequalities is proposed.