Thesis Summary:

The goal of this thesis is to develop hybridizable discontinuous Galerkin-type discretizations applied to non-linear elliptic problems from plasma physics. The complexity of these problems lies in the non-linearity of the unknowns and their source terms, as well as the fact that the equations are posed in non-polygonal domains. To deal with the curved boundaries, a high-order transfer technique is applied for the boundary data. First, we present the a priori and a posteriori error analysis of a high order hybridizable discontinuous Galerkin method (HDG) applied to a semi-linear elliptic problem, raised in a non- polygonal domain Ω. In this case, the non-linearity appears in the source term. We approximate Ω by a polygonal subdomain Ω_h and guarantee optimal convergence under mild assumptions related to the non-linear source term and the distance between the boundaries of the polygonal subdomain Ω_h and the original domain Ω. In addition, we use a local nonlinear post-processing of the scalar unknown to guarantee an additional order of convergence. Finally, we provide a reliable and locally efficient a posteriori error estimator that includes the approximation error between the original and artificial boundary data. Then, we extend the above analysis to a class of nonlinear elliptic boundary value problems posed on curved domains, where both the source term and the diffusion coefficient are nonlinear. The non-linearity of the diffusion coefficient can be presented by means of a scalar function or a vector function. Therefore, we divide the analysis into two cases: In the first one, we consider that the non-linear diffusion coefficient depends on the solution, while in the second case, this coefficient depends on the gradient of the solution. We also show that under minor assumptions about the source term and the computational domain, the discrete systems, for both cases, are well defined. In addition, we provide a priori error estimates that show that the discrete solution will have an optimal order of convergence as long as the distance between the curved boundary and the computational boundary remains the same order of magnitude as the mesh parameter. Finally, we propose a formulation that combines the HDG method with boundary element method (BEM) used for a more general problem from plasma physics. In this situation, the location of the plasma is unknown and it is necessary to solve the equilibrium condition in the half-plane to determine both the flow and the confinement region. The BEM method is ideal for working in unbounded domains, since the FEM approach would need an infinite number of elements to cover the domain.

ISI Publications from the Thesis

Nestor SáNCHEZ, Tonatiuh SANCHEZ-VIZUET, Manuel SOLANO: Afternote to Coupling at a distance: convergence analysis and a priori error estimates. Computational Methods in Applied Mathematics, vol. 22, no. 4, 2022, pp. 945 970, (2022).

Nestor SáNCHEZ, Tonatiuh SANCHEZ-VIZUET, Manuel SOLANO: Error analysis of an unfitted HDG method for a class of non-linear elliptic problems. Journal of Scientific Computing, vol. 90, 3, article: 92, (2022).

Nestor SáNCHEZ, Tonatiuh SANCHEZ-VIZUET, Manuel SOLANO: A priori and a posteriori error analysis of an unfitted HDG method for semi-linear elliptic problems. Numerische Mathematik, vol. 148, 4, pp. 919–958, (2021).

Other Publications (ISI)

Sergio CAUCAO, Gabriel N. GATICA, Ricardo OYARZúA, Nestor SáNCHEZ: A fully-mixed formulation for the steady double-diffusive convection system based upon Brinkman-Forchheimer equations. Journal of Scientific Computing, vol. 85, 2, article: 44, (2020).