# Residual minimization for isogeometric analysis in reduced and mixed forms

## TL;DR

This paper introduces a residual minimization framework for isogeometric analysis that uses highly-continuous basis functions for trial spaces and lower-regularity basis functions for test spaces, improving efficiency while maintaining accuracy.

## Contribution

It proposes a mixed formulation with residual minimization that allows using less regular test functions in isogeometric analysis, enhancing computational efficiency.

## Key findings

- Variationally-stable formulations verified numerically
- Equivalent formulations demonstrated through numerical tests
- Framework reduces regularity requirements for test spaces

## Abstract

Most variational forms of isogeometric analysis use highly-continuous basis functions for both trial and test spaces. For a partial differential equation with a smooth solution, isogeometric analysis with highly-continuous basis functions for trial space results in excellent discrete approximations of the solution. 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.