There has been considerable interest in recent years in solving multiphysics problems coupled through an interface. For example, geological reservoir models of flow in porous media often consist of multiple fault blocks which render a monolithic discretization infeasible. In such instances it is desirable to be able to discretize each subdomain (fault block) independently and produce a numerical approximation using whichever technique is most appropriate. Mortar domain decomposition methods provide a convenient and mathematically elegant approach for coupling different numerical methods through physically meaningful interface conditions. They do not require the grids to match along the interface and can easily be generalized to multiphase flow, multiphysics, computational mechanics, and geomechanics. The mortar mixed finite element method has also been shown to be equivalent to a multiscale method in the case where the mortar discretization is coarser than the subdomain discretizations. A significant portion of my PhD and postdoctoral research was devoted to the analysis of multiscale mortar methods with a particular focus on a posteriori error estimation and multiscale physics-based preconditioners.

**Relevant Publications:**

S. Tavener and T. Wildey,
**A Posteriori Error Estimates for Multiscale Mortar Discretizations**,
Accepted for publication in SIAM. J. Sci. Comput. 2013.

G. Pencheva, M. Vohralik, M.F. Wheeler, T. Wildey.
**Certified a posteriori error estimates for multiscale, multinumerics, and mortar
coupling**,
Pencheva, G., Vohralk, M., Wheeler, M., and Wildey, T. SIAM J. Numer. Anal. 51, 1 (2013), pp. 526-554.

V. Girault, G. Pencheva, M.F. Wheeler, T. Wildey.
**Domain decomposition for poroelasticity and elasticity with DG jumps and mortars**,
M3AS., 21 (2011) pp. 169-213.

M.F. Wheeler, T. Wildey, I. Yotov.
**A multiscale preconditioner for multiscale mortar mixed finite elements**.
Comp. Meth. in Appl. Mech. and Engng., 200 (2011) pp. 1251-1262.

M.F. Wheeler, T. Wildey, and G. Xue.
**Recent Advances in Multiscale Mortar Methods**,
Numerical Linear Algebra with Applications, Vol. 17, 2010, pp. 771-785.

D. Estep, S. Tavener, T. Wildey.
**A posteriori error estimation and adaptive mesh refinement for a multi-discretization operator decomposition approach to fluid-solid heat transfer**.
Journal of Computational Physics, 229 (2010), pp. 4143-4158.

V. Girault, G. Pencheva, M.F. Wheeler, T. Wildey.
**Domain decomposition for linear elasticity with DG jumps and mortars**.
Comp. Meth. in Appl. Mech. and Engng., 198 (2009) pp. 1751-1765.

D. Estep, S. Tavener, T. Wildey.
**A posteriori error analysis for a transient conjugate heat transfer problem**.
Finite Elements in Analysis and Design, 45 (2009) pp. 263-271.

D. Estep, S. Tavener, T. Wildey,
**A posteriori analysis and improved accuracy for an operator decomposition
solution of a conjugate heat transfer problem**,
SIAM J. Numer. Anal., 46 (2008), pp. 2068-2089.

Computational modeling is becoming increasingly reliant on a posterior error estimates to provide a measure of reliability on the numerical predictions. This methodology has been developed for a variety of methods and is widely accepted in the analysis of discretization error for partial differential equations. The adjoint-based (dual-weighted residual) method, is motivated by the observation that oftentimes the goal of a simulation is to compute a small number of linear functionals of the solution, such as the average value in a region or the drag on an object, rather than controlling the error in a global norm. Much of my research has been devoted to deriving a posteriori error estimates for different discretization methods, for operator decomposition approaches for multiscale/multiphysics applications, and for samples of surrogate models in uncertainty quantification.

**Relevant Publications:**

C. Bryant, S. Prudhomme, and T. Wildey,
**A posteriori Error Control for Partial Differential Equations with Random Data**,
Submitted to SIAM/ASA J. Uncert. Quant. 2013.

E. Cyr, J. Shadid, and T. Wildey.
**A posteriori Analysis of Stabilized Finite Element Methods**,
Submitted to SIAM J. Sci. Comput. 2012.

T. Wildey.
**A Posteriori Error Analysis of Interior Penalty Discontinuous Galerkin Methods**,
Submitted to Int. J. Numer. Model. Engrg. 2012.

S. Tavener and T. Wildey,
**A Posteriori Error Estimates for Multiscale Mortar Discretizations**,
Accepted for publication in SIAM. J. Sci. Comput. 2013.

T. Butler, C. Dawson, T. Wildey.
**Propagation of Uncertainties Using Improved Surrogate Model**,
Butler, T., Dawson, C., and Wildey, T. SIAM/ASA Journal on Uncertainty Quantification, 1 (2013), pp. 164-191.

G. Pencheva, M. Vohralik, M.F. Wheeler, T. Wildey.
**Certified a posteriori error estimates for multiscale, multinumerics, and mortar
coupling**,
Pencheva, G., Vohralk, M., Wheeler, M., and Wildey, T. SIAM J. Numer. Anal. 51, 1 (2013), pp. 526-554.

T. Butler, P. Constantine, T. Wildey.
**A Posteriori Error Analysis of Parameterized Linear Systems Using Spectral Methods**,
SIAM. J. Matrix Anal. Appl. 33, (2012) pp. 195-209.

T. Butler, C. Dawson, T. Wildey.
**A posteriori analysis of stochastic differential equation utilizing polynomial chaos expansions**,
SIAM J. Sci. Comput. 33, (2011) pp. 1267-1291.

D. Estep, S. Tavener, T. Wildey.
**A posteriori error estimation and adaptive mesh refinement for a multi-discretization operator decomposition approach to fluid-solid heat transfer**.
Journal of Computational Physics, 229 (2010), pp. 4143-4158.

D. Estep, S. Tavener, T. Wildey.
**A posteriori error analysis for a transient conjugate heat transfer problem**.
Finite Elements in Analysis and Design, 45 (2009) pp. 263-271.

D. Estep, V. Carey, V. Ginting, S. Tavener, T. Wildey,
**A posteriori error analysis of multiscale operator decomposition methods for multiphysics models**,
Journal of Physics: Conference Series 125 (2008), pp. 1-16.

D. Estep, S. Tavener, T. Wildey,
**A posteriori analysis and improved accuracy for an operator decomposition
solution of a conjugate heat transfer problem**,
SIAM J. Numer. Anal., 46 (2008), pp. 2068-2089.

T. Wildey, S. Tavener, and D. Estep,
**A posteriori error estimation of approximate boundary fluxes**,
Communications in Numerical Methods in Engineering, 24 (2008), pp. 421-434.

There is considerable interest in developing efficient and accurate methods to quantify the uncertainty in computational differential equation models. Often a two stage approach to solve this problem is formulated. First, a large number of samples of model parameters or input data are determined in terms of realizations of a stochastic process. Second, the probability distribution is approximately propagated through the computational model to the output or observable data. The first stage involves a priori knowledge and is often a modeling decision of the user, e.g. choosing to model a parameter as a random process with a uniform or Gaussian distribution. The majority of the computational burden is in the second stage involving the propagation of inputs to outputs. A simple approach is Monte Carlo simulation where the model is solved for each randomly generated input sample resulting in samples of the output distribution. From these output samples statistics or density estimates on specific quantities of interest may be computed. While the implementation is straightforward, this method can become computationally prohibitive for complex models where a large number of runs of the computational model may be infeasible. Further complicating the task of quantifying the uncertainty reliably is that each output sample is polluted by discretization error in the computational model. Moreover, the computational model may be extremely expensive to run, so only a limited number of samples can be computed and the statistical error will be quite large. Another approach for propagation of distributions is to construct a surrogate response surface and propagate samples using this surrogate rather than the full computational model. This approach essentially eliminates the statistical error component from the computed distribution since each sample of the surrogate model output has a very low computational cost implying a large number of samples may be taken. However, each of these samples may be contaminated by additional deterministic error from interpolating the surrogate approximation. Thus, the trade-off is that statistical error may be neglected at the cost of possibly large deterministic sources of error. A portion of my recent research has focused on estimating the deterministic error in samples of a surrogate model to improve the reliability and predictability of probabilistic quantities using computational models.

**Relevant Publications:**

C. Bryant, S. Prudhomme, and T. Wildey,
**A posteriori Error Control for Partial Differential Equations with Random Data**,
Submitted to SIAM/ASA J. Uncert. Quant. 2013.

P. Constantine, E. Phipps, and T. Wildey,
**Efficient uncertainty propagation for network multiphysics systems**,
Submitted to Int. J. Numer. Model. Engrg. 2013.

T. Butler, C. Dawson, T. Wildey.
**Propagation of Uncertainties Using Improved Surrogate Model**,
Butler, T., Dawson, C., and Wildey, T. SIAM/ASA Journal on Uncertainty Quantification, 1 (2013), pp. 164-191.

T. Butler, C. Dawson, T. Wildey.
**A posteriori analysis of stochastic differential equation utilizing polynomial chaos expansions**,
SIAM J. Sci. Comput. 33, (2011) pp. 1267-1291.

B. Ganis, H. Klie, M.F. Wheeler, T. Wildey, I. Yotov, and D. Zhang,
**Stochastic collocation and mixed finite elements for flow in porous media**,
Comp. Meth. in Appl. Mech. and Engng., 197 (2008) pp. 3547-3559.

Our ability to use computational models to make predictions is often predicated on the notion that we can (approximately) solve very large sparse linear systems in a reasonable amount of time. In most cases, this requires iterative solvers such as Krylov-based methods. Unfortunately, the performance of such methods is directly linked to the spectral properties of the linear system, i.e., the number of distinct eigenvalues. Ideally, a preconditioner is simply an easily invertible linear system with similar spectral properties to the original linear system. An active area of research aims to utilize multilevel or multiscale methods to efficiently propagate information throughout the domain of influence with the ultimate goal of improving the perfomance of the iterative solver.

**Relevant Publications:**

T. Wildey, G. Xue.
**Preconditioning for Mixed Finite Element Formulations of Elliptic Problems**,
in Domain Decomposition Methods in Science and Engineering XX, (2013), pp. 175-182, Springer.

B. Ganis, G. Pencheva, M.F. Wheeler, T. Wildey, I. Yotov.
**A frozen Jacobian multiscale preconditioner for nonlinear interface operators**,
Multiscale Model. Simul., 10(3), (2012) pp. 853-873.

M.F. Wheeler, T. Wildey, I. Yotov.
**A multiscale preconditioner for multiscale mortar mixed finite elements**.
Comp. Meth. in Appl. Mech. and Engng., 200 (2011) pp. 1251-1262.