We introduce the concept of optimal test functions that guarantee stability of resulting numerical schemes. Petrov-Galerkin methods seek approximate solutions of boundary value problems in a trial space by weakly imposing all equations via a (possibly different) test space. A basic design principle is that while trial spaces must have good approximation properties, the test space must be chosen for stability. The optimal test functions are those that realize discrete stability constants equal to those in the wellposedness estimates for the undiscretized boundary value problem. When such functions are used within an ultra-weak variational formulation, we obtain Discontinuous Petrov-Galerkin (DPG) methods that exhibit remarkable stability properties. We present the first complete theory for the DPG for Laplace’s equation as well as numerical results for other more complex applications.

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Last updated: 19 Jul 2024