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BIRS Workshop Lecture Videos

Combining h- and r-adaptivity for finite-element models with jumping coefficients Jahandari, Hormoz


Discontinuous (jumping) coefficients often appear in modelling problems where the computational domain represents inhomogeneous media. An example of this is the geophysical electromagnetic (EM) modelling problem, where these jumps occur at interfaces that separate regions with different conductivities. These interfaces, along with other problem features, such as singular EM sources and pointwise solution observations, motivate mesh refinement to achieve good accuracy. The goal of this study is to investigate the combined application of h- and r-refinement to reduce numerical error in the modelling of EM data. For simplicity, aspects of this hr-adaptivity are explored in 1D. The steady-state diffusion and Helmholtz equations (which are commonly solved for the EM scalar and vector potentials, respectively) constitute the physical PDEs (PPDEs) here, while the r-refinement is based on an equidistribution principle. The PPDEs and the mesh PDE are solved alternately in an iterative manner to reduce an error estimate to a desired level. At each iteration, the old mesh is h-refined, the error estimate and its corresponding monitor function are updated and the r-refinement is performed. Various finite-element (FE) a posteriori global and local error estimates were examined: while FE residual-based estimates were cheaper to compute, hierarchical error estimates were found to be better indicators of the true errors. The solutions of adjoint problems of the PPDEs were used to construct local error estimates. These estimates were successfully used for goal-oriented refinement of the FE models. The talk is based on a joint work with Scott MacLachlan and Ronald Haynes.

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