Residual Minimization for Isogeometric Analysis in Reduced and Mixed Forms

Most variational forms of isogeometric analysis use highly- continuous basis functions for both trial and test spaces. Isogeometric analysis results in excellent discrete approximations for differential equations with regular solutions. However, we observe that high continuity for test spaces is not necessary. In this work, we present a framework which uses highly-continuous B-splines for the trial spaces and basis functions with minimal regularity and possibly lower order polynomials for the test spaces. To realize this goal, we adopt the residual minimization methodology. We pose the problem in a mixed formulation, which results in a system governing both the solution and a Riesz representation of the residual. We present various variational formulations which are variationally-stable and verify their equivalence numerically via numerical tests.