Add more information to the README.

README.md

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 # HYPRE.jl
 
-[Julia][julia] interface to [HYPRE][hypre] (high performance preconditioners and solvers
+| **Documentation**         | **Build Status**                      |
+|:------------------------- |:------------------------------------- |
+| [![][docs-img]][docs-url] | [![][gh-actions-img]][gh-actions-url] |
+
+[Julia][julia] interface to [HYPRE][hypre] ("high performance preconditioners and solvers
 featuring multigrid methods for the solution of large, sparse linear systems of equations on
-massively parallel computers).
+massively parallel computers").
+
+While the main purpose of HYPRE is to solve problems on multiple cores, it can also be used
+for single core problems. HYPRE.jl aims to make it easy to use both modes of operation, with
+an interface that should be familiar to Julia programmers. This README includes some basic
+examples -- refer to the [documentation][docs-url] for more details, and for information
+about the included solvers and preconditioners and how to configure them.
+
+## Example: Single-core solve with standard sparse matrices
+
+It is possible to use Julia's standard sparse arrays (`SparseMatrixCSC` from the
+[SparseArrays.jl][sparse-stdlib] standard library, and `SparseMatrixCSR` from the
+[SparseMatricesCSR.jl][sparsecsr] package) directly in HYPRE.jl. For example, to solve
+`Ax = b` with conjugate gradients:
+
+```julia
+# Initialize linear system
+A = SparseMatrixCSC(...)
+b = Vector(...)
+
+# Create a conjugate gradients solver
+cg = HYPRE.PCG()
+
+# Compute the solution
+x = HYPRE.solve(cg, A, b)
+```
+
+## Example: Multi-core solve using PartitionedArrays.jl
+
+For multi-core problems it is possible to use [PartitionedArrays.jl][partarrays] directly
+with HYPRE.jl. Once the linear system is setup the solver interface is identical. For
+example, to solve `Ax = b` with bi-conjugate gradients and an algebraic multigrid
+preconditioner:
+
+```julia
+# Initialize linear system
+A = PSparseMatrix(...)
+b = PVector(...)
+
+# Create preconditioner
+precond = BoomerAMG()
+
+# Create a bi-conjugate gradients solver
+bicg = HYPRE.BiCGSTAB(; Precond = precond)
+
+# Compute the solution
+x = HYPRE.solve(bicg, A, b)
+```
+
 
 [julia]: https://julialang.org/
 [hypre]: https://github.com/hypre-space/hypre
+[sparse-stdlib]: https://github.com/JuliaSparse/SparseArrays.jl
+[sparsecsr]: https://github.com/gridap/SparseMatricesCSR.jl
+[partarrays]: https://github.com/fverdugo/PartitionedArrays.jl
+[docs-img]: https://img.shields.io/badge/docs-stable%20release-blue.svg
+[docs-url]: https://fredrikekre.github.io/HYPRE.jl/
+[gh-actions-img]: https://github.com/fredrikekre/HYPRE.jl/workflows/CI/badge.svg
+[gh-actions-url]: https://github.com/fredrikekre/HYPRE.jl/actions?query=workflow%3ACI