The Singularity Problem in Convection-Diffusion Models: A New Approach
In this article, we delve into the analysis and numerical results of a singular perturbed convection-diffusion problem and its discretization. Specifically, we focus on the scenario where the convection term dominates the problem, leading to interesting challenges in accurately approximating the solution.
Optimal Norm and Saddle Point Reformulation
One of the key contributions of our research is the introduction of the concept of optimal norm and saddle point reformulation in the context of mixed finite element methods. By utilizing these concepts, we were able to derive new error estimates specifically tailored for cases where the convection term is dominant.
These new error estimates provide valuable insights into the behavior of the numerical approximation and help us understand the limitations of traditional approaches. By comparing these estimates with those obtained from the standard linear Galerkin discretization, we gain a deeper understanding of the non-physical oscillations observed in the discrete solutions.
Saddle Point Least Square Discretization
In exploring alternative discretization techniques, we propose a novel approach called the saddle point least square discretization. This method utilizes quadratic test functions, which offers a more accurate representation of the solution compared to the linear Galerkin discretization.
Through our analysis, we shed light on the non-physical oscillations observed in the discrete solutions obtained using this method. Understanding the reasons behind these oscillations allows us to refine the discretization scheme and improve the accuracy of the numerical solution.
Relating Different Discretization Methods
In addition to our own proposed method, we also draw connections between other existing discretization methods commonly used for convection-diffusion problems. We emphasize the upwinding Petrov Galerkin method and the stream-line diffusion discretization method, highlighting their resulting linear systems and comparing the error norms associated with each.
By examining these relationships, we gain insights into the strengths and weaknesses of each method and can make informed decisions regarding their suitability for different scenarios. This comparative analysis allows us to choose the most efficient approximation technique for more general singular perturbed problems, including those with convection domination in multidimensional settings.
In conclusion, our research provides a comprehensive analysis of singular perturbed convection-diffusion problems, with a specific focus on cases dominated by the convection term. By introducing new error estimates, proposing a novel discretization method, and relating different approaches, we offer valuable insights into the numerical approximation of these problems. Our findings can be extended to tackle more complex and multidimensional scenarios, advancing the field of numerical approximation for singular perturbed problems.