Solving transient PDEs in Julia with Gridap.jl

In this talk we present a new feature of Gridap.jl focusing on the solution of transient Partial Differential Equations (PDEs). We will show a new API that: a) leads to weak forms with very simple syntax, b) supports automatic differentiation, c) enables the solution of multi-field and DAE systems, and d) can be used in parallel computing through GridapDistributed.jl. We will showcase the novel features for a variety of applications in fluid and solid dynamics.

Gridap is an open-source, finite element (FE) library implemented in the Julia programming language. The main goal of Gridap is to adopt a more modern programming style than existing FE applications written in C/C++ or Fortran in order to simplify the simulation of challenging problems in science and engineering and improve productivity in the research of new discretization methods. The library is a feature-rich general-purpose FE code able to solve a wide range of partial differential equations (PDEs), including linear, nonlinear, and multi-physics problems. Gridap is extensible and modular. One can implement new FE spaces, new reference elements, and use external mesh generators, linear solvers, and visualization tools. In addition, it blends perfectly well with other packages of the Julia package ecosystem, since Gridap is implemented 100% in Julia.

In this presentation we highlight a new feature introduced in Gridap.jl during the last year, a new high-level API that allows the user to simulate complex transient PDEs with very few lines of code. This new API goes in line with the distinctive features of Gridap.jl, allowing for the definition of weak forms in a syntax that is very similar to the mathematical notation used in academic works. The new API has a series of noticeable features, namely: it supports ODEs of arbitrary order, provided that a solver for the specific order is implemented, allows automatic differentiation of all the jacobians associated to the trannsient problem, enables the solution of multi-field and Diferential Algebraic Equation (DAE) systems, and can be used in parallel computing through the extension of the API to the GridapDistributed.jl package.

In JuliaCon2022 we will showcase this novel feature with a number of real applications in fluid and solid dynamics. The applications will include problems resulting in 1st and 2nd order ODEs, problems with constant and time-dependent coefficients, and problems with time-dependent geometries.