The Automatic Construction and Solution of a Partial Differential Equation from the Strong Form
In the last ten years, there has been significant improvement and growth in tools that aid the development of finite element methods for solving partial differential equations. These tools assist the user in transforming a weak form of a differential equation into a computable solution. Despite these advancements, solving a differential equation remains challenging. Not only are there many possible weak forms for a particular problem, but the most accurate or most efficient form depends on the problem’s structure. Requiring a user to generate a weak form by hand creates a significant hurdle for someone who understands a model, but does not know how to solve it.
We present a new algorithm that finds the solution of a partial differential equation when modeled in its strong form. We accomplish this by applying a first order system least squares algorithm using triangular Bézier patches as our shape functions. After describing our algorithm, we validate our results by presenting a numerical example.
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