Despite recent advances, there are still challenges in solving higher-order partial differential equations using the finite element method. This thesis focuses on finite element techniques for higher-order problems with applications to isogeometric analysis and statistical inference. In both areas, the efficient solution of higher-order partial differential equations is essential.

The construction of isogeometric smooth spline basis functions is crucial for solving higher-order partial differential equations on meshes with arbitrary topologies. We introduce a simple blending approach that produces smooth blended B-splines, referred to as SB-splines, on unstructured quadrilateral and hexahedral meshes. We first define a set of mixed smoothness quadratic B-splines that are $C0$ continuous in the unstructured regions of the mesh but are $C1$ continuous everywhere else. Subsequently, the SB-splines are obtained by smoothly blending in the physical space the mixed smoothness B-splines with Bernstein basis functions of equal degree. One of the key novelties of our approach is that the required smooth weight functions are assembled from the available smooth B-splines in the geometric parametrisation. The SB-splines are globally smooth, non-negative, have no breakpoints within the elements, and reduce to conventional B-splines away from the unstructured regions of the mesh. Remarkably, the optimal convergence rates are numerically observed for the Poisson and biharmonic problems in one and two dimensions.

The statistical finite element method coherently sythesises data and finite element models using Bayesian inference. However, the Gaussian process inference using covariance matrices is computationally expensive and memory-intensive. Consequently, the size of the finite element model is significantly limited. To this end, we introduce a sparse precision formulation by specifying the Gaussian process priors as Matérn random fields. Instead of kernel parametrisation, the Matérn random fields are parametrised as solutions of a higher-order stochastic partial differential equation. Finite element discretisation of the corresponding weak form results in sparse precision matrices. Specifically, the sparse precision matrices for arbitrary Matérn smoothness are obtained using recursive finite element computation and rational approximation. In addition to achieving improved scalability, we demonstrate the extension of the stochastic partial differential equation approach to non-Euclidean domains and the modelling of non-stationary and anisotropic Matérn random fields.