Solution Theory, Variational Formulations, and Functional a Posteriori Error Estimates for General First Order Systems with Applications to Electro-Magneto-Statics and More

We prove a comprehensive solution theory using tools from functional analysis, show corresponding variational formulations, and present functional a posteriori error estimates for general linear first order systems of type A2x=f,A1*x=g, for two densely defined and closed (possibly unbounded) linear operators A1 and A2 having the complex property A2A1=0. As a prototypical application we will discuss the system of electro-magneto statics in 3D with mixed tangential and normal boundary conditions rot E=F, − div εE=g. Our theory covers a lot more applications in 2D, 3D, and ND, such as general differential forms and all kind of systems arising, e.g., in general relativity, biharmonic problems, Stokes equations, or linear elasticity, to mention just a few, for example dE=F, RotSM=F, DivTT=F, Rot RotS⊤S=F, − δεE=G, divDivSεM=G, symRotTεT=G, − DivS εS=G, all with possibly mixed boundary conditions of generalized tangential and normal type. Second order systems of types A2*A2x=f, A2*A2x=f, A1*x=g, A1A1*x=g will be considered as well using the same techniques.