Block encoding for the finite element method with non-uniform mesh

Many quantum linear solvers reduce the solution of a discretized partial differential equation to repeated queries of a block encoding of the discretized operator. For the finite element method (FEM), efficient block encodings are so far available only on uniform meshes. One of the strength of finite elements is their use of graded meshes to resolve, e.g., corner singularities and boundary layers, which current quantum constructions cannot represent efficiently. We describe a block encoding of FEM Helmholtz operators that admits structured non-uniform meshes whose element-size sequence is set by a small number of parameters rather than one value per element, such as geometrically graded meshes. The depth of the encoding stays logarithmic in the number of elements and the subnormalization ratio is bounded by a constant.